#### Prof. Dr. Dmitri Kuzmin

Applied Mathematics

TU Dortmund University

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- kuzmin@math.uni-dortmund.de
- +49 0231 755 3461
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**A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws**

Kuzmin, D.*Computer Methods in Applied Mechanics and Engineering*373 (2021)In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a local discrete maximum principle (DMP), we impose inequality constraints on the local Lax–Friedrichs fluxes of a piecewise-linear (P1) approximation. Since the piecewise-constant (P0) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. After presenting two flux limiters that ensure the validity of local DMPs for cell averages, we discuss the design of slope limiters based on different kinds of inequality constraints. In particular, we derive new limiting procedures based on flux constraints and constraints for directional derivatives. The latter approach makes it possible to preserve directional monotonicity in applications to problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P1 subcells of macroelements marked as ‘troubled’ by a smoothness indicator. © 2020 Elsevier B.V.view abstract 10.1016/j.cma.2020.113569 **A positivity-preserving and conservative intersection-distribution-based remapping algorithm for staggered ALE hydrodynamics on arbitrary meshes**

Kenamond, M. and Kuzmin, D. and Shashkov, M.*Journal of Computational Physics*435 (2021)We introduce new intersection-distribution-based remapping tools for indirect staggered arbitrary Lagrangian-Eulerian (ALE) simulations of multi-material shock hydrodynamics on arbitrary meshes. In addition to conserving momentum and total energy, the three-stage remapper proposed in this work preserves non-negativity of the internal energy. At the first stage, we construct slope-limited piecewise-linear reconstructions of all conserved quantities on zones of the source mesh and perform intersection-based remap to obtain bound-preserving zonal quantities on the target mesh. At the second stage, we define bound-preserving nodal quantities of the staggered ALE discretization as convex combinations of corner quantities. The nodal internal energy is corrected in a way which keeps it non-negative, while providing exact conservation of total energy. At the final stage, we distribute the non-negative nodal internal energy to corners, zones and materials using non-negative weights. Proofs of positivity preservation are provided for each stage. This work is a natural extension of our paper [14] in which a similar intersection-distribution-based remapping procedure was employed. The original version used a nodal kinetic energy fix which did not provably ensure positivity preservation for the zonal internal energy after the final distribution stage. The new algorithm cures this potential drawback by using ‘coordinated’ limiters for piecewise-linear reconstructions, remapping the internal energy to nodes and correcting it before redistribution. The effectiveness of the new nodal fix is illustrated by numerical examples. © 2021 Elsevier Inc.view abstract 10.1016/j.jcp.2021.110254 **Intersection-distribution-based remapping between arbitrary meshes for staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics**

Kenamond, M. and Kuzmin, D. and Shashkov, M.*Journal of Computational Physics*429 (2021)We present a new intersection-distribution-based remapping method between arbitrary polygonal meshes for indirect staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics. All cell-centered material quantities are conservatively remapped using intersections between the Lagrangian (old, source) mesh and the rezoned (new, target) mesh. The new nodal masses are obtained by conservative distribution of all material masses in each new cell to the cell's corners and then collecting those corner masses at new nodes. This distribution is done using a local constrained optimization approach for each cell in the new mesh. In order to remap nodal momentum we first define cell-centered momentum for each cell in the old mesh, conservatively remap this to the new mesh and then conservatively distribute the new zonal momentum to each cell's bounding nodes, again using local constrained optimization. Our method also conserves total energy by applying a new nodal kinetic energy correction that relies on a process similar to that used for remapping nodal mass and momentum. Cell-centered kinetic energy is computed, conservatively remapped and then distributed to nodes. The discrepancy between this conservatively remapped and actual nodal kinetic energy is then conservatively distributed to the internal energies of the materials in the cells surrounding each node. Unlike conventional cell-based corrections of this type, this new nodal kinetic energy correction has not been observed to drive material internal energy negative in any of our testing. Unlike flux based remapping, our new intersection-distribution method can be applied to remapping between source and target meshes that are arbitrarily different, which provides superior flexibility in the rezoning strategy. Our method is accurate, essentially conservative and essentially bounds preserving. © 2020 Elsevier Inc.view abstract 10.1016/j.jcp.2020.110014 **A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations**

Mabuza, S. and Shadid, J.N. and Cyr, E.C. and Pawlowski, R.P. and Kuzmin, D.*Journal of Computational Physics*410 (2020)In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems. © 2020 Elsevier Inc.view abstract 10.1016/j.jcp.2020.109390 **Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws**

Kuzmin, D. and Quezada de Luna, M.*Computer Methods in Applied Mechanics and Engineering*372 (2020)In this work, we modify a continuous Galerkin discretization of a scalar hyperbolic conservation law using new algebraic correction procedures. Discrete entropy conditions are used to determine the minimal amount of entropy stabilization and constrain antidiffusive corrections of a property-preserving low-order scheme. The addition of a second-order entropy dissipative component to the antidiffusive part of a nearly entropy conservative numerical flux is generally insufficient to prevent violations of local bounds in shock regions. Our monolithic convex limiting technique adjusts a given target flux in a manner which guarantees preservation of invariant domains, validity of local maximum principles, and entropy stability. The new methodology combines the advantages of modern entropy stable/entropy conservative schemes and their local extremum diminishing counterparts. The process of algebraic flux correction is based on inequality constraints which provably provide the desired properties. No free parameters are involved. The proposed algebraic fixes are readily applicable to unstructured meshes, finite element methods, general time discretizations, and steady-state residuals. Numerical studies of explicit entropy-constrained schemes are performed for linear and nonlinear test problems. © 2020 Elsevier B.V.view abstract 10.1016/j.cma.2020.113370 **Bathymetry Reconstruction Using Inverse ShallowWater Models: Finite Element Discretization and Regularization**

Hajduk, H. and Kuzmin, D. and Aizinger, V.*Lecture Notes in Computational Science and Engineering*132 (2020)In the present paper, we use modified shallow water equations (SWE) to reconstruct the bottom topography (also called bathymetry) of a flow domain without resorting to traditional inverse modeling techniques such as adjoint methods. The discretization in space is performed using a piecewise linear discontinuous Galerkin (DG) approximation of the free surface elevation and (linear) continuous finite elements for the bathymetry. Our approach guarantees compatibility of the discrete forward and inverse problems: for a given DG solution of the forward SWE problem, the underlying continuous bathymetry can be recovered exactly. To ensure well-posedness of the modified SWE and reduce sensitivity of the results to noisy data, a regularization term is added to the equation for the water height. A numerical study is performed to demonstrate the ability of the proposed method to recover bathymetry in a robust and accurate manner. © Springer Nature Switzerland AG 2020.view abstract 10.1007/978-3-030-30705-9_20 **Bound-preserving flux limiting schemes for DG discretizations of conservation laws with applications to the Cahn–Hilliard equation**

Frank, F. and Rupp, A. and Kuzmin, D.*Computer Methods in Applied Mechanics and Engineering*359 (2020)Many mathematical models of computational fluid dynamics involve transport of conserved quantities, which must lie in a certain range to be physically meaningful. The analytical or numerical solution [Formula presented] of a scalar conservation law is said to be bound-preserving if global bounds u∗ and u∗ exist such that [Formula presented] holds in the domain of definition. These bounds must be known a priori. To enforce such inequality constraints at least for element averages in the context of discontinuous Galerkin (DG) methods, the numerical fluxes must be defined and constrained in an appropriate manner. In this work, we introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve relevant global bounds for element averages even if the analytical solution of the PDE violates them due to modeling errors. The proposed methodology is based on a combination of flux and slope limiting. The (optional) slope limiter adjusts the gradients to impose local bounds on pointwise values of the high-order DG solution, which is used to calculate the fluxes. The flux limiter constrains changes of element averages so as to prevent violations of global bounds. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. We propose two kinds of flux limiters, which meet this requirement. The first one is of monolithic type and its time-implicit version requires the iterative solution of a nonlinear problem. Only a fully converged solution is provably bound-preserving. The time-explicit version of this limiter is subject to a time step restriction, which we derive in this article. The second limiter is an iterative version of the multidimensional flux-corrected transport (FCT) algorithm and works as postprocessed correction scheme. This fractional step limiting approach guarantees that each iterate is bound-preserving but avoidable consistency errors may occur if the iterative process is terminated too early. Each iterate depends only on local information of the previous iterate. This concept of limiting the numerical fluxes is also applicable to finite volume methods. Practical applicability of the proposed flux limiters as well as the benefits of using an optional slope limiter are demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn–Hilliard equation (parabolic, nonlinear) for first-order polynomials. While both flux limiters work for arbitrary order polynomials, we discuss the construction of bound-preserving slope limiters, and show numerical studies only for first-order polynomials. © 2019 Elsevier B.V.view abstract 10.1016/j.cma.2019.112665 **Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws**

Kuzmin, D. and Quezada de Luna, M.*Computers and Fluids*213 (2020)This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems. © 2020 Elsevier Ltdview abstract 10.1016/j.compfluid.2020.104742 **Entropy stabilization and property-preserving limiters for P1 discontinuous Galerkin discretizations of scalar hyperbolic problems**

Kuzmin, D.*Journal of Numerical Mathematics*(2020)The methodology proposed in this paper bridges the gap between entropy stable and positivitypreserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions. © 2020 De Gruyter. All rights reserved.view abstract 10.1515/jnma-2020-0056 **Gradient-Based Limiting and Stabilization of Continuous Galerkin Methods**

Kuzmin, D.*Lecture Notes in Computational Science and Engineering*132 (2020)In this paper, we stabilize and limit continuous Galerkin discretizations of a linear transport equation using an algebraic approach to derivation of artificial diffusion operators. Building on recent advances in the analysis and design of edge-based algebraic flux correction schemes for singularly perturbed convection-diffusion problems, we derive algebraic stabilization operators that generate nonlinear high-order stabilization in smooth regions and enforce discrete maximum principles everywhere. The correction factors for antidiffusive element or edge contributions are defined in terms of nodal gradients that vanish at local extrema. The proposed limiting strategy is linearity-preserving and provides Lipschitz continuity of constrained terms. Numerical examples are presented for two-dimensional test problems. © Springer Nature Switzerland AG 2020.view abstract 10.1007/978-3-030-30705-9_29 **Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations**

Kuzmin, D. and Klyushnev, N.*Journal of Computational Physics*407 (2020)This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks. © 2020 Elsevier Inc.view abstract 10.1016/j.jcp.2020.109230 **Locally bound-preserving enriched Galerkin methods for the linear advection equation**

Kuzmin, D. and Hajduk, H. and Rupp, A.*Computers and Fluids*205 (2020)In this work, we introduce algebraic flux correction schemes for enriched (P1⊕P0 and Q1⊕P0) Galerkin discretizations of the linear advection equation. The piecewise-constant component stabilizes the continuous Galerkin approximation without introducing free parameters. However, violations of discrete maximum principles are possible in the neighborhood of discontinuities and steep fronts. To keep the cell averages and the degrees of freedom of the continuous P1/Q1 component in the admissible range, we limit the fluxes and element contributions, the complete removal of which would correspond to first-order upwinding. The first limiting procedure that we consider in this paper is based on the flux-corrected transport (FCT) paradigm. It belongs to the family of predictor-corrector algorithms and requires the use of small time steps. The second limiting strategy is monolithic and produces nonlinear problems with well-defined residuals. This kind of limiting is well suited for stationary and time-dependent problems alike. The need for inverting consistent mass matrices in explicit strong stability preserving Runge–Kutta time integrators is avoided by reconstructing nodal time derivatives from cell averages. Numerical studies are performed for standard 2D test problems. © 2020 Elsevier Ltdview abstract 10.1016/j.compfluid.2020.104525 **Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations**

Hajduk, H. and Kuzmin, D. and Kolev, T. and Abgrall, R.*Computer Methods in Applied Mechanics and Engineering*359 (2020)In this work, we introduce a new residual distribution (RD) framework for the design of bound-preserving high-resolution finite element schemes. The continuous and discontinuous Galerkin discretizations of the linear advection equation are modified to construct local extremum diminishing (LED) approximations. To that end, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high-order Bernstein finite element discretizations involves localization to subcells and definition of bound-preserving weights for subcell contributions. Using strong stability preserving (SSP) Runge–Kutta methods for time integration, we prove the validity of discrete maximum principles under CFL-like time step restrictions. The low-order version of our method has roughly the same accuracy as the one derived from a piecewise (multi)-linear approximation on a submesh with the same nodal points. In high-order extensions, we use an element-based flux-corrected transport (FCT) algorithm which can be interpreted as a nonlinear RD scheme. The proposed LED corrections are tailor-made for matrix-free implementations which avoid the rapidly growing cost of matrix assembly for high-order Bernstein elements. The results for 1D, 2D, and 3D test problems compare favorably to those obtained with the best matrix-based approaches. © 2019 Elsevier B.V.view abstract 10.1016/j.cma.2019.112658 **Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting**

Hajduk, H. and Kuzmin, D. and Kolev, T. and Tomov, V. and Tomas, I. and Shadid, J.N.*Computers and Fluids*200 (2020)This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems. © 2020 Elsevier Ltdview abstract 10.1016/j.compfluid.2020.104451 **Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws**

Kuzmin, D.*Computer Methods in Applied Mechanics and Engineering*361 (2020)Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to flux-corrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions. Each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds. There is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately. Moreover, the steady-state limit of the nonlinear discrete problem is well defined and independent of the pseudo-time step. In the case study for the Euler equations, the components of the bar states are constrained sequentially to satisfy local maximum principles for the density, velocity, and specific total energy in addition to positivity preservation for the density and pressure. The results of numerical experiments for standard test problems illustrate the ability of built-in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner. © 2019 Elsevier B.V.view abstract 10.1016/j.cma.2019.112804 **Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere**

Novikov, A. and Kuzmin, D. and Ahmadi, O.*Applied Mathematics and Computation*364 (2020)This paper is focused on efficient Monte Carlo simulations of Brownian diffusion effects in particle-based numerical methods for solving transport equations on a sphere (or a circle). Using the heat equation as a model problem, random walks are designed to emulate the action of the Laplace–Beltrami operator without evolving or reconstructing the probability density function. The intensity of perturbations is fitted to the value of the rotary diffusion coefficient in the deterministic model. Simplified forms of Brownian motion generators are derived for rotated reference frames, and several practical approaches to generating random walks on a sphere are discussed. The alternatives considered in this work include projections of Cartesian random walks, as well as polar random walks on the tangential plane. In addition, we explore the possibility of using look-up tables for the exact cumulative probability of perturbations. Numerical studies are performed to assess the practical utility of the methods under investigation. © 2019 Elsevier Inc.view abstract 10.1016/j.amc.2019.124670 **Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws**

Kuzmin, D. and Quezada de Luna, M.*Journal of Computational Physics*411 (2020)This work extends the concepts of algebraic flux correction and convex limiting to continuous high-order Bernstein finite element discretizations of scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the standard Galerkin approximation is transformed into a nonlinear high-resolution scheme which has the compact sparsity pattern of the piecewise-linear or multilinear subcell discretization. The representation of this scheme in terms of invariant domain preserving states makes it possible to prove the validity of local discrete maximum principles under CFL-like conditions. In contrast to predictor-corrector approaches based on the flux-corrected transport methodology, the proposed flux limiting strategy is monolithic, i.e., limited antidiffusive terms are incorporated into the well-defined residual of a nonlinear (semi-)discrete problem. A stabilized high-order Galerkin discretization is recovered if no limiting is performed. In the limited version, the compact stencil property prevents direct mass exchange between nodes that are not nearest neighbors. A formal proof of sparsity is provided for simplicial and box elements. The involved element contributions can be calculated efficiently making use of matrix-free algorithms and precomputed element matrices of the reference element. Numerical studies for Q2 discretizations of linear and nonlinear two-dimensional test problems illustrate the virtues of monolithic convex limiting based on subcell flux decompositions. © 2020 Elsevier Inc.view abstract 10.1016/j.jcp.2020.109411 **A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds**

Sokolov, A. and Davydov, O. and Kuzmin, D. and Westermann, A. and Turek, S.*Journal of Numerical Mathematics*27 (2019)In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical behavior of the method by performing numerical tests for the solid-body rotation benchmark in a unit square and for a transport problem along a curve implicitly prescribed by a level set function. Extension of the proposed method to higher dimensions is straightforward and easily realizable. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.view abstract 10.1515/jnma-2018-0097 **A monolithic conservative level set method with built-in redistancing**

Quezada de Luna, M. and Kuzmin, D. and Kees, C.E.*Journal of Computational Physics*379 (2019)We introduce a new level set method for representing evolving interfaces. In the case of divergence-free velocity fields, the new method satisfies a conservation principle. Conservation is important for many applications such as modeling two-phase incompressible flow. In the present implementation, the conserved quantity is defined as the integral of a smoothed characteristic function. The new approach embeds level sets into a volume of fluid formulation. The evolution of an approximate signed distance function is governed by a conservation law for its (smoothed) sign. The non-linear level set transport equation is regularized by adding a flux correction term that assures a non-singular Jacobian and penalizes deviations from a distance function. The result is a locally conservative level set method with built-in elliptic redistancing. The continuous model is monolithic in the sense that the level set transport model, the volume of fluid law of mass conservation, and the minimization problem that preserves the approximate distance function property are incorporated into a single equation. There is no need for any extra stabilization, artificial compression, flux limiting, redistancing, mass correction, and other numerical fixes which are commonly used in level set or volume of fluid methods. In addition, there is just one free parameter that controls the strength of regularization and penalization in the model. The accuracy and conservation properties of the monolithic finite element/level set method are illustrated by the results of numerical studies for passive advection of free interfaces. © 2018view abstract 10.1016/j.jcp.2018.11.044 **A partition of unity approach to adaptivity and limiting in continuous finite element methods**

Kuzmin, D. and Quezada de Luna, M. and Kees, C.E.*Computers and Mathematics with Applications*78 (2019)The partition of unity finite element method (PUFEM) proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner. The lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures. In contrast to traditional hp-adaptivity for finite elements, the proposed approach preserves discrete conservation properties and the continuity of traces at common boundaries of adjacent mesh cells. In the context of space discretizations, a continuous blending function is used to combine finite element bases corresponding to high-order polynomials and piecewise-linear approximations based on the same set of nodes. In a similar vein, spatially partitioned time discretizations can be designed using weights that depend continuously on the space variable. The design of blending functions may be based on a priori knowledge (e.g., in applications to problems with singularities or boundary layers), local error estimates, smoothness indicators, and/or discrete maximum principles. In adaptive methods, changes of the finite element approximation exhibit continuous dependence on the data. The presented numerical examples illustrate the typical behavior of local H1 and L2 errors. © 2019 Elsevier Ltdview abstract 10.1016/j.camwa.2019.03.021 **An updated Lagrangian discontinuous Galerkin hydrodynamic method for gas dynamics**

Wu, T. and Shashkov, M. and Morgan, N. and Kuzmin, D. and Luo, H.*Computers and Mathematics with Applications*78 (2019)We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for gas dynamics. The new method evolves conserved unknowns in the current configuration, which obviates the Jacobi matrix that maps the element in a reference coordinate system or the initial coordinate system to the current configuration. The density, momentum, and total energy (ρ,ρu,E) are approximated with conservative higher-order Taylor expansions over the element and are limited toward a piecewise constant field near discontinuities using a limiter. Two new limiting methods are presented for enforcing the bounds on the primitive variables of density, velocity, and specific internal energy (ρ,u,e). The nodal velocity, and the corresponding forces, are calculated by solving an approximate Riemann problem at the element nodes. An explicit second-order method is used to temporally advance the solution. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. 1D Cartesian coordinates test problem results are presented to demonstrate the accuracy and convergence order of the new DG method with the new limiters. © 2018 Elsevier Ltdview abstract 10.1016/j.camwa.2018.03.040 **New directional vector limiters for discontinuous Galerkin methods**

Hajduk, H. and Kuzmin, D. and Aizinger, V.*Journal of Computational Physics*384 (2019)Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertex-based slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics. © 2019 Elsevier Inc.view abstract 10.1016/j.jcp.2019.01.032 **Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems**

Mabuza, S. and Shadid, J.N. and Kuzmin, D.*Journal of Computational Physics*361 (2018)The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized. © 2018view abstract 10.1016/j.jcp.2018.01.048 **Planar and orthotropic closures for orientation tensors in fiber suspension flow models**

Kuzmin, D.*SIAM Journal on Applied Mathematics*78 (2018)This paper presents a bottom-up approach to derivation of orientation tensor closures for fiber suspension flow models. To begin with, we consider polynomial approximations based on the two-dimensional (2D) versions of the linear, quadratic, natural, and orthotropic smooth closures for reconstruction of the fourth-order orientation tensor. A numerical study is performed for simple flows. The investigation of planar closures provides new insights and boundary conditions for the design of orthotropic closures in three dimensions. The proposed extensions use finite element shape functions to interpolate the data at principal orientation states and additional points. The results for 3D simple flows indicate that natural closures based on (extended) quadratic and piecewise-linear interpolation provide a far better description of the 3D orientation dynamics than any other orthotropic closure considered in this study. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/18M1175665 **Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations**

Dobrev, V. and Kolev, T. and Kuzmin, D. and Rieben, R. and Tomov, V.*Journal of Computational Physics*356 (2018)We present a new predictor-corrector approach to enforcing local maximum principles in piecewise-linear finite element schemes for the compressible Euler equations. The new element-based limiting strategy is suitable for continuous and discontinuous Galerkin methods alike. In contrast to synchronized limiting techniques for systems of conservation laws, we constrain the density, momentum, and total energy in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum gradients are adjusted to incorporate the irreversible effect of density changes. Antidiffusive corrections to bounds-compatible low-order approximations are limited to satisfy inequality constraints for the specific total and kinetic energy. An accuracy-preserving smoothness indicator is introduced to gradually adjust lower bounds for the element-based correction factors. The employed smoothness criterion is based on a Hessian determinant test for the density. A numerical study is performed for test problems with smooth and discontinuous solutions. © 2017 Elsevier Inc.view abstract 10.1016/j.jcp.2017.12.012 **An FCT finite element scheme for ideal MHD equations in 1D and 2D**

Basting, M. and Kuzmin, D.*Journal of Computational Physics*338 (2017)This paper presents an implicit finite element (FE) scheme for solving the equations of ideal magnetohydrodynamics in 1D and 2D. The continuous Galerkin approximation is constrained using a flux-corrected transport (FCT) algorithm. The underlying low-order scheme is constructed using a Rusanov-type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed. The accuracy of the low-order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity-preserving iterative manner. At the FCT correction step, the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure. Divergence-free magnetic fields are extracted using projections of the FCT predictor into staggered finite element spaces forming exact sequences. In the 2D case, the magnetic field is projected into the space of Raviart–Thomas finite elements. Numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal MHD flows. © 2017 Elsevier Inc.view abstract 10.1016/j.jcp.2017.02.051 **Anisotropic slope limiting for discontinuous Galerkin methods**

Aizinger, V. and Kosík, A. and Kuzmin, D. and Reuter, B.*International Journal for Numerical Methods in Fluids*84 (2017)In this paper, we present an anisotropic version of a vertex-based slope limiter for discontinuous Galerkin methods. The limiting procedure is carried out locally on each mesh element utilizing the bounds defined at each vertex by the largest and smallest mean value from all elements containing the vertex. The application of this slope limiter guarantees the preservation of monotonicity. Unnecessary limiting of smooth directional derivatives is prevented by constraining the x and y components of the gradient separately. As an inexpensive alternative to optimization-based methods based on solving small linear programming problems, we propose a simple operator splitting technique for calculating the correction factors for the x and y derivatives. We also provide the necessary generalizations for using the anisotropic limiting strategy in an arbitrary rotated frame of reference and in the vicinity of exterior boundaries with no Dirichlet information. The limiting procedure can be extended to elements of arbitrary polygonal shape and three dimensions in a straightforward fashion. The performance of the new anisotropic slope limiter is illustrated by two-dimensional numerical examples that employ piecewise linear discontinuous Galerkin approximations. © 2017 John Wiley & Sons, Ltd.view abstract 10.1002/fld.4360 **Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements**

Lohmann, C. and Kuzmin, D. and Shadid, J.N. and Mabuza, S.*Journal of Computational Physics*344 (2017)This work extends the flux-corrected transport (FCT) methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier net. The design of accuracy-preserving FCT schemes for high order Bernstein–Bézier finite elements requires the development of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils, (ii) high order variational stabilization based on the difference between two gradient approximations, and (iii) new localized limiting techniques for antidiffusive element contributions. The optional use of a smoothness indicator, based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D. © 2017 Elsevier Inc.view abstract 10.1016/j.jcp.2017.04.059 **Gradient-based nodal limiters for artificial diffusion operators in finite element schemes for transport equations**

Kuzmin, D. and Shadid, J.N.*International Journal for Numerical Methods in Fluids*(2017)This paper presents new linearity-preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high-order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum dim inishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation that implies the preservation of second-order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions. © 2017 John Wiley & Sons, Ltd.view abstract 10.1002/fld.4365 **High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation**

Anderson, R. and Dobrev, V. and Kolev, T. and Kuzmin, D. and Quezada de Luna, M. and Rieben, R. and Tomov, V.*Journal of Computational Physics*334 (2017)In this work we present a FCT-like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported. © 2016 Elsevier Inc.view abstract 10.1016/j.jcp.2016.12.031 **Linearity-preserving monotone local projection stabilization schemes for continuous finite elements**

Kuzmin, D. and Basting, S. and Shadid, J.N.*Computer Methods in Applied Mechanics and Engineering*322 (2017)This paper presents some new tools for enforcing discrete maximum principles and/or positivity preservation in continuous piecewise-linear finite element approximations to convection-dominated transport problems. Using a linear first-order advection equation as a model problem, we construct element-level bilinear forms associated with first-order artificial diffusion operators and their two-scale counterparts. The underlying design philosophy is similar to that behind local projection stabilization (LPS) techniques and variational multiscale (VMS) methods. The difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected. The proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation. The value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots. The result is a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The use of variational gradient recovery makes it possible to add high-order background dissipation leading to improved approximation properties in smooth regions. The numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time-dependent test problems in two space dimensions. © 2017 Elsevier B.V.view abstract 10.1016/j.cma.2017.04.030 **Embedded discontinuous Galerkin transport schemes with localised limiters**

Cotter, C.J. and Kuzmin, D.*Journal of Computational Physics*311 (2016)Motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction, we introduce locally bounded transport schemes for (partially-)continuous finite element spaces. The underlying high-order transport scheme is constructed by injecting the partially-continuous field into an embedding discontinuous finite element space, applying a stable upwind discontinuous Galerkin (DG) scheme, and projecting back into the partially-continuous space; we call this an embedded DG transport scheme. We prove that this scheme is stable in L2 provided that the underlying upwind DG scheme is. We then provide a framework for applying limiters for embedded DG transport schemes. Standard DG limiters are applied during the underlying DG scheme. We introduce a new localised form of element-based flux-correction which we apply to limiting the projection back into the partially-continuous space, so that the whole transport scheme is bounded. We provide details in the specific case of tensor-product finite element spaces on wedge elements that are discontinuous P1/Q1 in the horizontal and continuous P2 in the vertical. The framework is illustrated with numerical tests. © 2016 Elsevier Inc.view abstract 10.1016/j.jcp.2016.02.021 **Optimal control for reinitialization in finite element level set methods**

Basting, C. and Kuzmin, D. and Shadid, J.N.*International Journal for Numerical Methods in Fluids*84 (2016)A new optimal control problem that incorporates the residual of the Eikonal equation into its objective is presented. The formulation of the state equation is based on the level set transport equation but extended by an additional source term, correcting the solution so as to minimize the objective functional. The method enforces the constraint so that the interface cannot be displaced at least in the continuous setting. The system of first-order optimality conditions is derived, linearized, and solved numerically. The control also prevents numerical instabilities, so that no additional stabilization techniques are required. This approach offers the flexibility to include other desired design criteria into the objective functional. The methodology is evaluated numerically in three different examples and compared with other PDE-based reinitialization techniques. © 2016 John Wiley & Sons, Ltd.view abstract 10.1002/fld.4348 **Synchronized flux limiting for gas dynamics variables**

Lohmann, C. and Kuzmin, D.*Journal of Computational Physics*326 (2016)This work addresses the design of failsafe flux limiters for systems of conserved quantities and derived variables in numerical schemes for the equations of gas dynamics. Building on Zalesak's multidimensional flux-corrected transport (FCT) algorithm, we construct a new positivity-preserving limiter for the density, total energy, and pressure. The bounds for the underlying inequality constraints are designed to enforce local maximum principles in regions of strong density variations and become less restrictive in smooth regions. The proposed approach leads to closed-form expressions for the synchronized correction factors without the need to solve inequality-constrained optimization problems. A numerical study is performed for the compressible Euler equations discretized using a finite element based FCT scheme. © 2016 Elsevier Inc.view abstract 10.1016/j.jcp.2016.09.025 **Analysis of a combined CG1-DG2 method for the transport equation**

Becker, R. and Bittl, M. and Kuzmin, D.*SIAM Journal on Numerical Analysis*53 (2015)In this paper, we introduce a reduced discontinuous Galerkin method in which the space of continuous piecewise-linear functions (CG1) is enriched with discontinuous piecewisequadratics (DG2). The resultant finite element approximation is continuous at the vertices of the mesh and discontinuous across edges/faces. We analyze the properties of the CG1-DG2 discretization in the context of a steady linear transport equation. The presented a priori error estimate shows that the discontinuous enrichment stabilizes the continuous coarse-scale component and delivers optimal convergence rates. Numerical studies for steady and unsteady convection problems confirm this result. © 2015 Society for Industrial and Applied Mathematics.view abstract 10.1137/13093683X **Scale separation in fast hierarchical solvers for discontinuous Galerkin methods**

Aizinger, V. and Kuzmin, D. and Korous, L.*Applied Mathematics and Computation*266 (2015)We present a method for solution of linear systems resulting from discontinuous Galerkin (DG) approximations. The two-level algorithm is based on a hierarchical scale separation scheme (HSS) such that the linear system is solved globally only for the cell mean values which represent the coarse scales of the DG solution. The system matrix of this coarse-scale problem is exactly the same as in the cell-centered finite volume method. The higher order components of the solution (fine scales) are computed as corrections by solving small local problems. This technique is particularly efficient for DG schemes that employ hierarchical bases and leads to an unconditionally stable method for stationary and time-dependent hyperbolic and parabolic problems. Unlike p-multigrid schemes, only two levels are used for DG approximations of any order. The proposed method is conceptually simple and easy to implement. It compares favorably to p-multigrid in our numerical experiments. Numerical tests confirm the accuracy and robustness of the proposed algorithm. © 2015 Elsevier Inc.view abstract 10.1016/j.amc.2015.05.047 **A conservative, positivity preserving scheme for reactive solute transport problems in moving domains**

Mabuza, S. and Kuzmin, D. and Čanić, S. and Bukač, M.*Journal of Computational Physics*276 (2014)We study the mathematical models and numerical schemes for reactive transport of a soluble substance in deformable media. The medium is a channel with compliant adsorbing walls. The solutes are dissolved in the fluid flowing through the channel. The fluid, which carries the solutes, is viscous and incompressible. The reactive process is described as a general physico-chemical process taking place on the compliant channel wall. The problem is modeled by a convection-diffusion adsorption-desorption equation in moving domains. We present a conservative, positivity preserving, high resolution ALE-FCT scheme for this problem in the presence of dominant transport processes and wall reactions on the moving wall. A Patankar type time discretization is presented, which provides conservative treatment of nonlinear reactive terms. We establish CFL-type constraints on the time step, and show the mass conservation of the time discretization scheme. Numerical simulations are performed to show validity of the schemes against effective models under various scenarios including linear adsorption-desorption, irreversible wall reaction, infinite adsorption kinetics, and nonlinear Langmuir kinetics. The grid convergence of the numerical scheme is studied for the case of fixed meshes and moving meshes in fixed domains. Finally, we simulate reactive transport in moving domains under linear and nonlinear chemical reactions at the wall, and show that the motion of the compliant channel wall enhances adsorption of the solute from the fluid to the channel wall. Consequences of this result are significant in the area of, e.g., nano-particle cancer drug delivery. Our result shows that periodic excitation of the cancerous tissue using, e.g., ultrasound, may enhance adsorption of cancer drugs carried by nano-particles via the human vasculature. © 2014 Elsevier Inc.view abstract 10.1016/j.jcp.2014.07.049 **A nonlinear ALE-FCT scheme for non-equilibrium reactive solute transport in moving domains**

Mabuza, S. and Kuzmin, D.*International Journal for Numerical Methods in Fluids*76 (2014)In this paper, we present a conservative, positivity-preserving, high-resolution nonlinear ALE-flux-corrected transport (FCT) scheme for reactive transport models in moving domains. The mathematical model is a convection-diffusion equation with a nonlinear flux equation on the moving channel wall. The reactive transport is assumed to have dominant Peclet and Damkohler numbers, a phenomenon that often results in non-physical negative solutions. The scheme presented here is proven to be mass conservative in time and positive at all times for a small enough t. Reactive transport examples are simulated using this scheme for its validation, to show its convergence, and to compare it against the linear ALE-FCT scheme. The nonlinear ALE-FCT is shown to perform better than the linear ALE-FCT schemes for large time steps. Copyright (c) 2014 John Wiley & Sons, Ltd.view abstract 10.1002/fld.3961 **An optimization-based approach to enforcing mass conservation in level set methods**

Kuzmin, D.*Journal of Computational and Applied Mathematics*258 (2014)This paper presents a new conservative level set method for numerical simulation of evolving interfaces. A PDE-constrained optimization problem is formulated and solved in an iterative fashion. The proposed optimal control procedure constrains the level set function to satisfy a conservation law for the corresponding Heaviside function. The target value of the state variable is defined as the solution to the standard level set transport equation. The gradient of the control variable corrects the convective flux in the nonlinear state equation so as to enforce mass conservation while minimizing deviations from the target state. A relaxation term is added when it comes to the design of an iterative solver for the nonlinear system. The potential of the optimization-based approach is illustrated by two numerical examples. © 2013 Published by Elsevier B.V.view abstract 10.1016/j.cam.2013.09.009 **Finite element simulation of three-dimensional particulate flows using mixture models**

Gorb, Y. and Mierka, O. and Rivkind, L. and Kuzmin, D.*Journal of Computational and Applied Mathematics*270 (2014)In this paper, we discuss the numerical treatment of three-dimensional mixture models for (semi-)dilute and concentrated suspensions of particles in incompressible fluids. The generalized Navier-Stokes system and the continuity equation for the volume fraction of the disperse phase are discretized using an implicit high-resolution finite element scheme, and maximum principles are enforced using algebraic flux correction. To prevent the volume fractions from exceeding the maximum packing limit, a conservative overshoot limiter is applied to the converged convective fluxes at the end of each time step. A numerical study of the proposed approach is performed for 3D particulate flows over a backward-facing step and in a lid-driven cavity. © 2013 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2013.12.020 **Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods**

Kuzmin, D.*Journal of Computational Physics*257 (2014)In this paper, we present a collection of algorithmic tools for constraining high-order discontinuous Galerkin (DG) approximations to hyperbolic conservation laws. We begin with a review of hierarchical slope limiting techniques for explicit DG methods. A new interpretation of these techniques leads to an unconditionally stable implicit algorithm for steady-state computations. The implicit global problem for the mean values (coarse scales) has the computational structure of a finite volume method. The constrained derivatives (fine scales) are obtained by solving small local problems. The interscale transfer operators provide a two-way iterative coupling between the solutions to the global and local problems. Another highlight of this paper is a new approach to compatible gradient limiting for the Euler equations of gas dynamics. After limiting the conserved quantities, the gradients of the velocity and energy density are constrained in a consistent manner. Numerical studies confirm the accuracy and robustness of the proposed algorithms. © 2013 Elsevier Inc.view abstract 10.1016/j.jcp.2013.04.032 **Optimal control for mass conservative level set methods**

Basting, C. and Kuzmin, D.*Journal of Computational and Applied Mathematics*270 (2014)This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach. © 2013 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2013.12.040 **The CG1-DG2 method for convection-diffusion equations in 2D**

Bittl, M. and Kuzmin, D. and Becker, R.*Journal of Computational and Applied Mathematics*270 (2014)In this paper, we present the CG1-DG2 method for convection-diffusion equations. The space of continuous piecewise-linear functions is enriched with discontinuous quadratics so that the resultant finite element approximation is continuous at the vertices of the mesh but may have jumps across the edges. Three different approaches to the discretization of the diffusive part are considered: the symmetric interior penalty Galerkin method, the non-symmetric interior penalty Galerkin method and the Baumann-Oden method. In the context of elliptic problems we summarize well-known a priori error estimates for the discontinuous Galerkin approximation which carry over to the CG1-DG2 approach. Both methods have the same convergence rate which is also confirmed by numerical studies for diffusion and convection-diffusion problems. © 2014 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2014.03.008 **The reference solution approach to hp-adaptivity in finite element flux-corrected transport algorithms**

Bittl, M. and Kuzmin, D.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*8353 LNCS (2014)This paper presents an hp-adaptive flux-corrected transport algorithm based on the reference solution approach. It features a finite element approximation with unconstrained high-order elements in smooth regions and constrained Q1 elements in the neighborhood of steep fronts. The difference between the reference solution and its projection into the current (coarse) space is used as an error indicator to determine the local mesh size h and polynomial degree p. The reference space is created by increasing the polynomial degree p in smooth elements and h-refining the mesh in nonsmooth elements. The smoothness is determined by a hierarchical regularity estimator based on discontinuous higher-order reconstructions of the solution and its derivatives. The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) scheme. p-refinement is performed by enriching a continuous bilinear approximation with continuous or discontinuous basis functions of polynomial degree p ≥ 2. The algorithm is implemented in the open-source software package HERMES. The use of hierarchical data structures that support arbitrary level hanging nodes makes the extension of FCT to hp-FEM relatively straightforward. The accuracy of the proposed methodology is illustrated by a numerical example for a two-dimensional benchmark problem with a known exact solution. © 2014 Springer-Verlag.view abstract 10.1007/978-3-662-43880-0_21 **A minimization-based finite element formulation for interface-preserving level set reinitialization**

Basting, C. and Kuzmin, D.*Computing*95 (2013)This paper presents a new approach to reinitialization in finite element methods for the level set transport equation. The proposed variational formulation is derived by solving a minimization problem. A penalty term is introduced to preserve the shape of the free interface in the process of redistancing. In contrast to hyperbolic PDE reinitialization, the resulting boundary value problem is elliptic and can be solved using a simple fixed-point iteration method. The minimization-based approach makes it possible to define the desired geometric properties in terms of a suitable potential function. In particular, truncated distance functions can be generated using a doublewell potential. The results of a numerical study indicate that the new methodology is a promising alternative to conventional reinitialization techniques. © 2013 Springer-Verlag Wien.view abstract 10.1007/s00607-012-0259-z **A parameter-free smoothness indicator for high-resolution finite element schemes**

Kuzmin, D. and Schieweck, F.*Central European Journal of Mathematics*11 (2013)This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere. © 2013 Versita Warsaw and Springer-Verlag Wien.view abstract 10.2478/s11533-013-0254-4 **A positivity-preserving finite element method for chemotaxis problems in 3D**

Strehl, R. and Sokolov, A. and Kuzmin, D. and Horstmann, D. and Turek, S.*Journal of Computational and Applied Mathematics*239 (2013)We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2012.09.041 **An hp-adaptive flux-corrected transport algorithm for continuous finite elements**

Bittl, M. and Kuzmin, D.*Computing*95 (2013)This paper presents an hp-adaptive flux-corrected transport algorithm for continuous finite elements. The proposed approach is based on a continuous Galerkin approximation with unconstrained higher-order elements in smooth regions and constrained P1/Q1 elements in the neighborhood of steep fronts. Smooth elements are found using a hierarchical smoothness indicator based on discontinuous higher-order reconstructions. A gradient-based error indicator determines the local mesh size h and polynomial degree p. The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) algorithm. The same limiting strategy is employed when it comes to constraining the L2 projection of data from one finite-dimensional space into another. The new algorithm is implemented in the open-source software package Hermes. The use of hierarchical data structures that support arbitrary-level hanging nodes makes the extension of FCT to hp-FEM relatively straightforward. The accuracy of the proposed method is illustrated by a numerical study for a two-dimensional benchmark problem with a known exact solution. © 2013 Springer-Verlag Wien.view abstract 10.1007/s00607-012-0223-y **Numerical study of a high order 3D FEM-level set approach for immiscible flow simulation**

Turek, S. and Mierka, O. and Hysing, S. and Kuzmin, D.*Computational Methods in Applied Sciences*27 (2013)Numerical simulation of incompressible multiphase flows with immiscible fluids is still a challenging field, particularly for 3D configurations undergoing complex topological changes. In this paper, we discuss a 3D FEM approach with high-order Stokes elements (Q2/P1) for velocity and pressure on general hexahedral meshes. A discontinuous Galerkin approach with piecewise linear polynomials (dG(1)) is used to treat the Level Set function. The developed methodology allows the application of special redistancing algorithms which do not change the position of the interface. We explain the corresponding FEM techniques for treating the advection steps and surface tension effects, and validate the corresponding 3D code with respect to both numerical test cases and experimental data. The corresponding applications describe the classical rising bubble problem for various parameters and the generation of droplets from a viscous liquid jet in a coflowing surrounding fluid. Both of these applications can be used for rigorous benchmarking of 3D multiphase flow simulations. © 2013 Springer Science+Business Media Dordrecht.view abstract 10.1007/978-94-007-5288-7_4 **Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis**

Kuzmin, D.*International Journal for Numerical Methods in Fluids*71 (2013)The use of high-order polynomials in discontinuous Galerkin (DG) approximations to convection-dominated transport problems tends to cause a violation of the maximum principle in regions where the derivatives of the solution are large. In this paper, we express the DG solution in terms of Taylor basis functions associated with the cell average and derivatives at the center of the cell. To control the (derivatives of the) discontinuous solution, the values at the vertices of each element are required to be bounded by the means. This constraint is enforced using a hierarchical vertex-based slope limiter to constrain the coefficients of the Taylor polynomial in a conservative manner starting with the highest-order terms. The loss of accuracy at smooth extrema is avoided by taking the maximum of the correction factors for derivatives of order p and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. In the case of a non-orthogonal Taylor basis, the same limiter is applied to the vector of discretized time derivatives before the multiplication by the off-diagonal part of the consistent mass matrix. This strategy leads to a remarkable gain of accuracy, especially in the case of simplex meshes. A numerical study is performed for a 2D convection equation discretized with linear and quadratic finite elements. © 2012 John Wiley & Sons, Ltd.view abstract 10.1002/fld.3707 **A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions**

Kuzmin, D. and Gorb, Y.*Journal of Computational and Applied Mathematics*236 (2012)Convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level. In this paper, we are concerned with solving continuity equations that govern the evolution of volume fractions in Eulerian models of disperse two-phase flows. An implicit Galerkin finite element approximation is equipped with a flux limiter for the convective terms. The fully multidimensional limiting strategy is based on a flux-corrected transport (FCT) algorithm. This nonlinear high-resolution scheme satisfies a discrete maximum principle for divergence-free velocities and ensures positivity preservation for arbitrary velocity fields. To enforce an upper bound that corresponds to the maximum-packing limit, an FCT-like overshoot limiter is applied to the converged convective fluxes at the end of each time step. This postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model. Numerical results for 2D implosion problems illustrate the performance of the proposed limiting procedure. © 2011 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2011.10.019 **Implicit finite element schemes for the stationary compressible Euler equations**

Gurris, M. and Kuzmin, D. and Turek, S.*International Journal for Numerical Methods in Fluids*69 (2012)A semi-implicit finite element scheme and a Newton-like solver are developed for the stationary compressible Euler equations. Since the Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable, the troublesome antidiffusive part is constrained within the framework of algebraic flux correction. A generalization of total variation diminishing (TVD) schemes is employed to blend the original Galerkin scheme with its nonoscillatory low-order counterpart. Unlike standard TVD limiters, the proposed limiting strategy is fully multidimensional and readily applicable to unstructured meshes. However, the nonlinearity and nondifferentiability of the limiter function makes efficient computation of stationary solutions a highly challenging task, especially in situations when the Mach number is large in some subdomains and small in other subdomains. In this paper, a semi-implicit scheme is derived via a time-lagged linearization of the Jacobian operator, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability, as well as higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. The overall spatial accuracy of the constrained scheme and the benefits of the new boundary treatment are illustrated by grid convergence studies for 2D benchmark problems. © 2011 John Wiley & Sons, Ltd.view abstract 10.1002/fld.2532 **Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes**

Kuzmin, D.*Journal of Computational and Applied Mathematics*236 (2012)This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix. The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D. © 2011 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2011.11.019 **A Newton-like finite element scheme for compressible gas flows**

Gurris, M. and Kuzmin, D. and Turek, S.*Computers and Fluids*46 (2011)Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense. © 2011 Elsevier Ltd.view abstract 10.1016/j.compfluid.2011.01.025 **A positivity-preserving ALE finite element scheme for convection-diffusion equations in moving domains**

Boiarkine, O. and Kuzmin, D. and Čanić, S. and Guidoboni, G. and Mikelić, A.*Journal of Computational Physics*230 (2011)A new high-resolution scheme is developed for convection-diffusion problems in domains with moving boundaries. A finite element approximation of the governing equation is designed within the framework of a conservative Arbitrary Lagrangian Eulerian (ALE) formulation. An implicit flux-corrected transport (FCT) algorithm is implemented to suppress spurious undershoots and overshoots appearing in convection-dominated problems. A detailed numerical study is performed for P 1 finite element discretizations on fixed and moving meshes. Simulation results for a Taylor dispersion problem (moderate Peclet numbers) and for a convection-dominated problem (large Peclet numbers) are presented to give a flavor of practical applications. © 2011 Elsevier Inc.view abstract 10.1016/j.jcp.2010.12.042 **Erratum: Particle behavior during the arc spraying process with cored wires (Journal of Thermal Spray Technology DOI: 10.1007/s11666-008-9272-3)**

Tillmann, W. and Vogli, E. and Abdulgader, M. and Gurris, M. and Kuzmin, D. and Turek, S.*Journal of Thermal Spray Technology*20 (2011)view abstract 10.1007/s11666-008-9288-8 **Implicit finite element schemes for stationary compressible particle-laden gas flows**

Gurris, M. and Kuzmin, D. and Turek, S.*Journal of Computational and Applied Mathematics*235 (2011)The derivation of macroscopic models for particle-laden gas flows is reviewed. Semi-implicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust. © 2011 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2011.04.036 **Numerical aspects and implementation of population balance equations coupled with turbulent fluid dynamics**

Bayraktar, E. and Mierka, O. and Platte, F. and Kuzmin, D. and Turek, S.*Computers and Chemical Engineering*35 (2011)In this paper, we present numerical techniques for one-way coupling of CFD and Population Balance Equations (PBE) based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k-e{open} turbulence model, and breakage and coalescence closures. The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers wide range of computational and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through a static mixer. The aim of this paper is to highlight the influence of different formulations of the novel theoretical breakage and coalescence models on the equilibrium distribution of population, and to propose an implementation strategy for three-dimensional one-way coupled CFD-PBE model. © 2011 Elsevier Ltd.view abstract 10.1016/j.compchemeng.2011.04.001 **A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods**

Kuzmin, D.*Journal of Computational and Applied Mathematics*233 (2010)A new approach to slope limiting for discontinuous Galerkin methods on arbitrary meshes is introduced. A local Taylor basis is employed to express the approximate solution in terms of cell averages and derivatives at cell centroids. In contrast to traditional slope limiting techniques, the upper and lower bounds for admissible variations are defined using the maxima/minima of centroid values over the set of elements meeting at a vertex. The correction factors are determined by a vertex-based counterpart of the Barth-Jespersen limiter. The coefficients in the Taylor series expansion are limited in a hierarchical manner, starting with the highest-order derivatives. The loss of accuracy at smooth extrema is avoided by taking the maximum of correction factors for derivatives of order p ≥ 1 and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. Numerical examples are presented for 2D transport problems discretized using a DG method. © 2009 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2009.05.028 **Failsafe flux limiting and constrained data projections for equations of gas dynamics**

Kuzmin, D. and Möller, M. and Shadid, J.N. and Shashkov, M.*Journal of Computational Physics*229 (2010)A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L2 projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples. © 2010 Elsevier Inc.view abstract 10.1016/j.jcp.2010.08.009 **Finite element simulation of compressible particle-laden gas flows**

Gurris, M. and Kuzmin, D. and Turek, S.*Journal of Computational and Applied Mathematics*233 (2010)A macroscopic two-fluid model of compressible particle-laden gas flows is considered. The governing equations are discretized by a high-resolution finite element method based on algebraic flux correction. A multidimensional limiter of TVD type is employed to constrain the local characteristic variables for the continuous gas phase and conservative fluxes for a suspension of solid particles. Special emphasis is laid on the efficient computation of steady state solutions at arbitrary Mach numbers. To avoid stability restrictions and convergence problems, the characteristic boundary conditions are imposed weakly and treated in a fully implicit manner. A two-way coupling via the interphase drag force is implemented using operator splitting. The Douglas-Rachford scheme is found to provide a robust treatment of the interphase exchange terms within the framework of a fractional-step solution strategy. Two-dimensional simulation results are presented for a moving shock wave and for a steady nozzle flow. © 2009 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2009.07.041 **Goal-oriented a posteriori error estimates for transport problems**

Kuzmin, D. and Korotov, S.*Mathematics and Computers in Simulation*80 (2010)Some aspects of goal-oriented a posteriori error estimation are addressed in the context of steady convection-diffusion equations. The difference between the exact and approximate values of a linear target functional is expressed in terms of integrals that depend on the solutions to the primal and dual problems. Gradient averaging techniques are employed to separate the element residual and diffusive flux errors without introducing jump terms. The dual solution is computed numerically and interpolated using higher-order basis functions. A node-based approach to localization of global errors in the quantities of interest is pursued. A possible violation of Galerkin orthogonality is taken into account. Numerical experiments are performed for centered and upwind-biased approximations of a 1D boundary value problem. © 2009 IMACS.view abstract 10.1016/j.matcom.2009.03.008 **Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems**

Kuzmin, D. and Möller, M.*Journal of Computational and Applied Mathematics*233 (2010)The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems. The Galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints. The localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions. A nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible. The developed simulation tools are applied to a linear convection problem in two space dimensions. © 2009 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cam.2009.07.026

#### finite element method

#### numerical methods