On Entropy Consistency of Large Time Step Schemes I. The Godunov and Glimm Schemes

Wang, Jinghua; Warnecke, Gerald

doi:10.1137/0730064

Cited by 8 publications

(5 citation statements)

References 20 publications

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“…At the same time, in 1993, Wang and Warnecke investigated the entropy consistency in different LTS methods [144,145]. In 1995, further applications of the LTS methods included front tracking methods based on wave propagation by LeVeque and Shyue [86] and semi-implicit methods by Klein [68].…”

Section: Large Timementioning

confidence: 99%

“…Contributions to this matter have been made by Wang and Warnecke [144,145] in the nineteen nineties, where they proved that the LTS-Godunov and LTS-Glimm schemes are entropy stable forC ≤ 2 if the flux function is monotone, and for an arbitrary Courant number if the initial data is monotone. Later, Wang et al [146] proved entropy stability of the LTS-Godunov method for any Courant number for some additional types of initial data.…”

Section: Entropy Stabilitymentioning

confidence: 99%

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Large Time Step Roe scheme for a common 1D two-fluid model

Prebeg

Flåtten

Müller

2017

Applied Mathematical Modelling

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Large Time Step (LTS) finite volume methods are presented, analyzed and applied to one-dimensional hyperbolic conservation laws.The HLL (Harten-Lax-van Leer) and HLLC (HLL-Contact) schemes are extended to LTS-HLL(C) schemes. The LTS-HLL-type schemes for scalar conservation laws are defined in the numerical viscosity, flux-difference splitting and wave propagation form, and TVD conditions on wave velocity estimates are determined. It is shown that the LTS-HLL-type schemes contain some already existing LTS methods such as the LTS-Roe and LTS-Lax-Friedrichs, and that they allow us to deduce LTS extensions of other methods, such as the Rusanov and Engquist-Osher schemes. The LTS-HLL(C) schemes for systems of conservation laws are developed by standard field-by-field decomposition. Numerical results suggest that the computational efficiency of LTS methods depends on the type of problem under consideration. In certain cases LTS methods for nonlinear systems of equations yield increased accuracy, efficiency and convergence rate compared to standard methods.Entropy stability of LTS-HLL-type schemes is analyzed. We use modified equation analysis to gain insight into numerical diffusion in LTS-HLLtype schemes and to conjecture about entropy stability of LTS methods. Theoretical results obtained through the modified equation analysis are in agreement with numerical results for both scalar conservation laws and the Euler equations.Positivity preservation in LTS methods is investigated. We show that the positivity preserving conditions for LTS methods are stronger than for standard methods. We describe different ways positivity can be lost in LTS-HLL-type schemes for the Euler equations, and we propose a simple way to increase the robustness of the LTS-HLL-type schemes by adding numerical diffusion. Numerical investigations show that LTS-HLL-type schemes successfully handle test cases for positivity preservation, and where the positivity is lost we can improve the robustness by adding numerical diffusion.In addition, we applied the LTS-Roe scheme to a two-fluid model and focused on the treatment of the boundary conditions and the source terms. It is shown that stability and accuracy of the solution can be greatly improved by appropriate treatment of boundary conditions and source terms.i

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Section: Large Timementioning

confidence: 99%

Section: Entropy Stabilitymentioning

confidence: 99%

Large Time Step Roe scheme for a common 1D two-fluid model

Prebeg

Flåtten

Müller

2017

Applied Mathematical Modelling

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“…Using the Roe approximate Riemann solver as building block, similar results were obtained by Brenier [1]. The entropy consistency of these schemes was studied by Wang and Warnecke [14,15].…”

Section: A Description Of the Lts Schemementioning

confidence: 54%

“…7a,b. (14) also holds in this case. Summing up (11), (12) and (14), it can be concluded that, for Case 3, we can choose α, β and ω such that…”

Section: Lemma 5 For Case 3 We Can Choose α β and ω Such Thatmentioning

confidence: 61%

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Error bounds for the large time step Glimm scheme applied to scalar conservation laws

Huang

Wang

Warnecke³

2002

Numerische Mathematik

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In this paper we derive an L 1 error bound for the large time step, i.e. large Courant number, version of the Glimm scheme when used for the approximation of solutions to a genuinely nonlinear, i.e. convex, scalar conservation law for a generic class of piecewise constant data. We show that the error is bounded by O(∆x 1/2 | log ∆x|) for Courant numbers up to 1. The order of the error is the same as that given by Hoff and Smoller [5] in 1985 for the Glimm scheme under the restriction of Courant numbers up to 1/2. Classification (1991): 65M15, 35L65 Mathematics Subject

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The Case of Multidimensional Systems

Godlewski¹,

Raviart²

2020

Applied Mathematical Sciences

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On Entropy Consistency of Large Time Step Schemes I. The Godunov and Glimm Schemes

Cited by 8 publications

References 20 publications

Large Time Step Roe scheme for a common 1D two-fluid model

Large Time Step Roe scheme for a common 1D two-fluid model

Error bounds for the large time step Glimm scheme applied to scalar conservation laws

The Case of Multidimensional Systems

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