An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

Jiang, Yi; Liu, Hailiang

doi:10.1002/num.22274

Cited by 8 publications

(8 citation statements)

References 34 publications

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“…[10,11,8,12,13]) developed the IRP approximations in the context of continuous finite elements with convex limiting for solving general hyperbolic systems including the compressible Euler equations. Jiang and Liu proposed new IRP limiters for the DG schemes to the isentropic Euler equations [20], the compressible Euler equations [17], and general multi-dimensional hyperbolic conservation laws [18]. Gouasmi et al [6] proved a minimum entropy principle on entropy solutions to the compressible multicomponent Euler equations at the smooth and discrete levels.…”

Section: Introductionmentioning

confidence: 99%

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Wu¹

2021

Preprint

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This paper first explores Tadmor's minimum entropy principle for the special relativistic hydrodynamics (RHD) equations and incorporates this principle into the design of robust highorder discontinuous Galerkin (DG) and finite volume schemes for RHD on general meshes. The proposed schemes are rigorously proven to preserve numerical solutions in a global invariant region constituted by all the known intrinsic constraints: minimum entropy principle, the subluminal constraint on fluid velocity, the positivity of pressure, and the positivity of restmass density. Relativistic effects lead to some essential difficulties in the present study, which are not encountered in the non-relativistic case. Most notably, in the RHD case the specific entropy is a highly nonlinear implicit function of the conservative variables, and, moreover, there is also no explicit formula of the flux in terms of the conservative variables. In order to overcome the resulting challenges, we first propose a novel equivalent form of the invariant region, by skillfully introducing two auxiliary variables. As a notable feature, all the constraints in the novel form are explicit and linear with respect to the conservative variables. This provides a highly effective approach to theoretically analyze the invariant-region-preserving (IRP) property of numerical schemes for RHD, without any assumption on the IRP property of the exact Riemann solver. Based on this, we prove the convexity of the invariant region and establish the generalized Lax-Friedrichs splitting properties via technical estimates, lying the foundation for our rigorous IRP analyses. It is rigorously shown that the first-order Lax-Friedrichs type scheme for the RHD equations satisfies a local minimum entropy principle and is IRP under a CFL condition. Provably IRP high-order accurate DG and finite volume methods are then developed for the RHD with the help of a simple scaling limiter, which is designed by following the bound-preserving type limiters in the literature. Several numerical examples demonstrate the effectiveness and robustness of the proposed schemes.

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Section: Introductionmentioning

confidence: 99%

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Wu¹

2021

Preprint

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“…While the bound preserving limiter [33] for the entropy function was shown to be high order accurate provided the second order derivative of the entropy function for numerical solutions does not vanish. The work [15] was built upon [14], where we introduced an explicit IRP limiter for DG methods solving the isentropic gas dynamic system (with or without viscosity). Again both convexity and concavity of two Riemann invariants play an essential role in the construction of the explicit IRP limiter there.…”

Section: Introductionmentioning

confidence: 99%

“…The work [15] was built upon [14], where we introduced an explicit IRP limiter for DG methods solving the isentropic gas dynamic system (with or without viscosity). Again both convexity and concavity of two Riemann invariants play an essential role in the construction of the explicit IRP limiter there.…”

Section: Introductionmentioning

confidence: 99%

Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations

Jiang

Liu

2018

Journal of Computational Physics

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An invariant-region-preserving (IRP) limiter for multi-dimensional hyperbolic conservation law systems is introduced, as long as the system admits a global invariant region which is a convex set in the phase space. It is shown that the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average is away from the boundary of the convex set. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to hyperbolic conservation law systems, sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one-dimensional system shares the same invariant region as the full multidimensional hyperbolic system does. The general results are then applied to both one and two dimensional compressible Euler equations so to obtain high order IRP DG schemes. Numerical experiments are provided to validate the proven properties of the IRP limiter and the performance of IRP DG schemes for compressible Euler equations.

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“…The limiting techniques of this nature are inspired by the limiter introduced for conservation laws [43]; see also invariant region preserving limiters for systems of conservation laws in [15,16]. We now investigate the admissible parameters (β 0 , β 1 ) of the DDG scheme for the Poisson problem.…”

Section: 2mentioning

confidence: 99%

Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

Liu,

Wang,

Yin

et al. 2021

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In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson-Nernst-Planck (PNP) equations, which has found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

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An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

Cited by 8 publications

References 34 publications

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Minimum Principle on Specific Entropy and High-Order Accurate Invariant Region Preserving Numerical Methods for Relativistic Hydrodynamics

Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations

Positivity-preserving third order DG schemes for Poisson--Nernst--Planck equations

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