On large time step Godunov scheme for hyperbolic conservation laws

Wang, Jinghua; Wen, Hairui; Zhou, Tie

doi:10.4310/cms.2004.v2.n3.a7

Cited by 4 publications

(8 citation statements)

References 13 publications

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“…From 2001 to 2004, there was a focus on investigating more mathematical properties of LTS methods: Helzel and Warnecke [55] and Wang et al [146] studied the entropy stability of the LTS-Godunov scheme, Huang et al [62] investigated the error bounds for the LTS-Glimm scheme and Tang and Warnecke [133] studied the monotonicity of (2K+1)-point schemes.…”

Section: Large Timementioning

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%

“…Later, Wang et al [146] proved entropy stability of the LTS-Godunov method for any Courant number for some additional types of initial data. An example of monotone, hence entropy stable LTS method is an LTS version of Lax-Friedrichs scheme studied by Tang and Warnecke [133].…”

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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“…A parallel large time step version of the Glimm scheme and similar results were subsequently given by LeVeque [9], Brenier [1] and Wang [12,13]. In addition, Wang [12] obtained the consistency of the LTS Glimm scheme for systems of conservation laws in one space dimension for any choice of time step and convergence of the scheme for the case that C ≤ 1, if the total variation of v 0 is sufficiently small and each eigenvalue of the system does not change its' sign. Using the Roe approximate Riemann solver as building block, similar results were obtained by Brenier [1].…”

Section: A Description Of the Lts Schemementioning

confidence: 74%

“…(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: 98%

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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“…settingû j+1/2 (x, t) to be the solution to the Riemann problem (35) for (79) yields a conservative approximation to Godunov's method. For scalar equations, this is also known as Murman's method [18,24].…”

Section: The Large Timementioning

confidence: 99%

Large Time Step TVD Schemes for Hyperbolic Conservation Laws

Lindqvist¹,

Aursand²,

Flåtten³

et al. 2016

SIAM J. Numer. Anal.

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Abstract. Large time step explicit schemes in the form originally proposed by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463-477] have seen a significant revival in recent years. In this paper we consider a general framework of local 2k + 1 point schemes containing LeVeque's scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient.We identify the least and most diffusive TVD schemes in this framework. The most diffusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The least diffusive scheme is the Large Time Step scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes.We discuss the nature of entropy violations associated with the LTS-Roe scheme, in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions, and discuss numerical entropy fixes.Finally, we propose a one-parameter family of Large TimeStep TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The 1D Burgers equation and the Euler system are used as numerical illustrations.

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On large time step Godunov scheme for hyperbolic conservation laws

Cited by 4 publications

References 13 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

Large Time Step TVD Schemes for Hyperbolic Conservation Laws

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