Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

Kuzmin, Dmitri; Luna, Manuel Quezada de; Ketcheson, David I.; Grüll, Johanna

doi:10.1007/s10915-022-01784-0

Cited by 14 publications

(7 citation statements)

References 45 publications

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“…The experimental order of convergence (EOC) is above 2.0 if no limiting is performed and approximately 1.75 otherwise. Further improvements can be achieved by using less restrictive bounds, as proposed in [37]. The remarkably similar convergence behavior of the FC-L and DC schemes and the larger absolute errors of the SC method indicate that the choice of the local bounds has a stronger impact on the accuracy of slope-limited approximations to smooth solutions than the way in which these bounds are enforced in our methods.…”

Section: Steady Circular Advectionmentioning

confidence: 85%

“…To preserve the high accuracy and DMP property of the space discretization, time integration may need to be performed using a flux-corrected Runge-Kutta method of sufficiently high order. The first representatives of such methods were recently developed in [37,42]. In summary, the proposed methodology can be extended to high-order space-time discretizations but many additional aspects must be taken into account to reap the potential benefits.…”

Section: Discussionmentioning

confidence: 99%

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A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

Kuzmin

2020

Preprint

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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 (P 1 ) approximation. Since the piecewise-constant (P 0 ) 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. Moreover, both fluxes and slopes can be limited in a manner which produces nonlinear problems with welldefined residuals even at steady state. We derive/present slope limiters based on different kinds of inequality constraints, discuss their properties and introduce new anisotropic limiters for 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-P 1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P 1 subcells of macroelements marked as 'troubled' by a smoothness indicator.

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Section: Steady Circular Advectionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

Kuzmin

2020

Preprint

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“…A fundamental mathematical property of any reasonable numerical scheme for nonlinear systems of hyperbolic conservation laws that was not investigated in this paper at all was the invariant domain preserving property (IDP), such as the positivity of density and entropy. Future work will consider provably IDP extensions of our methods, making use of the mathematical techniques presented in the seminal work of Guermond and Popov et al concerning provably invariant domain preserving schemes [31,[57][58][59] and in the work of Kuzmin et al [7,[65][66][67] concerning bound-preserving algebraic flux limiters and slope limiters for high order continuous and discontinuous Galerkin finite element schemes.…”

Section: Discussionmentioning

confidence: 99%

A New Family of Thermodynamically Compatible Discontinuous Galerkin Methods for Continuum Mechanics and Turbulent Shallow Water Flows

Busto

Dumbser

2022

J Sci Comput

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In this work we propose a new family of high order accurate semi-discrete discontinuous Galerkin (DG) finite element schemes for the thermodynamically compatible discretization of overdetermined first order hyperbolic systems. In particular, we consider a first order hyperbolic model of turbulent shallow water flows, as well as the unified first order hyperbolic Godunov–Peshkov–Romenski (GPR) model of continuum mechanics, which is able to describe at the same time viscous fluids and nonlinear elastic solids at large deformations. Both PDE systems treated in this paper belong to the class of hyperbolic and thermodynamically compatible systems, since both satisfy an entropy inequality and the total energy conservation can be obtained as a direct consequence of all other governing equations via suitable linear combination with the corresponding thermodynamic dual variables. In this paper, we mimic this process for the first time also at the semi-discrete level at the aid of high order discontinuous Galerkin finite element schemes. For the GPR model we directly discretize the entropy inequality and obtain total energy conservation as a consequence of a suitable discretization of all other evolution equations. For turbulent shallow water flows we directly discretize the nonconservative evolution equations related to the Reynolds stress tensor and obtain total energy conservation again as a result of the thermodynamically compatible discretization. As a consequence, for continuum mechanics the new DG schemes satisfy a cell entropy inequality directly by construction and thanks to the discrete thermodynamic compatibility they are provably nonlinearly stable in the energy norm for both systems under consideration.

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“…In what follows, we propose ω = ω(r) so that this property is fulfilled for C ≤ 1. For larger Courant numbers, to preserve the TVD property, we have to consider l i ∈ [0, 1], see later their definition inspired by flux-corrected type methods [22,11].…”

Section: Linear Advection Equationmentioning

confidence: 99%

“…In the case of discontinuous solutions we propose a predictor-corrector procedure to find solution dependent values of the parameter for which the scheme is Total Variation Diminishing (TVD). For the case of Courant number larger than one, in general, additional limiting must be used, which we propose in a form of flux-corrected transport scheme [11,22]. The system of algebraic equations is solved efficiently with only one forward and one backward sweep using the fast sweeping method [25].…”

Section: Introductionmentioning

confidence: 99%

Semi-implicit high resolution numerical scheme for conservation laws

Frolkovič,

Žeravý

2022

Preprint

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We present a novel semi-implicit scheme for numerical solutions of time-dependent conservation laws. The core idea of the presented method consists of exploiting and approximating mixed partial derivatives of the solution that occur naturally when deriving higher-order accurate schemes. Such an approach is introduced in the context of the Lax-Wendroff (or Cauchy-Kowalevski) procedure when the time derivatives are not completely replaced by space derivatives using the PDE, but some mixed derivatives are allowed. If approximated in a suitable way, one obtains algebraic systems that have a more convenient structure than the systems derived by standard fully implicit schemes. We derive high-resolution TVD form of the semi-implicit scheme for some representative hyperbolic equations in one-dimensional case including related numerical experiments.

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Bound-preserving Flux Limiting for High-Order Explicit Runge–Kutta Time Discretizations of Hyperbolic Conservation Laws

Cited by 14 publications

References 45 publications

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

A New Family of Thermodynamically Compatible Discontinuous Galerkin Methods for Continuum Mechanics and Turbulent Shallow Water Flows

Semi-implicit high resolution numerical scheme for conservation laws

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