A discontinuous wave-in-cell numerical scheme for hyperbolic conservation laws

Thompson, Richard J.; Moeller, Trevor M.

doi:10.1016/j.jcp.2015.06.047

Cited by 6 publications

(3 citation statements)

References 29 publications

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“…17]), or to integrate the calculation of source terms directly with the wave propagation. This work verifies that the LTS Roe scheme, already successfully applied to shallow water flow [17,19], aerodynamics [22] and Maxwell's equations [26], can be extended to more general equation systems.…”

Section: Discussionsupporting

confidence: 65%

A Large Time Step Roe Scheme Applied to Two-Phase Flow

Lindqvist¹,

Lund²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We consider a large time step (LTS) Roe scheme, originally suggested by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463-477] which has attained increased popularity in recent years. The scheme is based on a wave formulation of the ordinary Roe scheme, and the assumption that waves interact linearly. This assumption lets us propagate waves over more than one cell, thereby violating the ordinary CFL condition requiring the Courant number to be less than one. The LTS Roe scheme is applied to a two-phase flow model with phase transfer, modelled using the stiffened-gas equation of state. Results from various test cases show that the LTS Roe scheme with moderate Courant numbers is significantly more efficient than the ordinary Roe scheme, measured by an error-to-runtime ratio. The optimal Courant number is mainly a trade-off between reduced runtime due to fewer time steps, and increased error due to the assumption of linear wave interactions.

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Section: Discussionsupporting

confidence: 65%

A Large Time Step Roe Scheme Applied to Two-Phase Flow

Lindqvist¹,

Lund²

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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“…The interested reader is referred to the research on semi-Lagrangian schemes [44,126,41,49,48,75], front-tracking methods studied by Holden and co-workers [56,58,57], different approaches by Corrias et al [20,21], Leonard [76,77,78], Qiu and Shu [119] and Thompson and Moeller [135], and a version of LTS method developed by Harten [51] and further explored by Qian and Lee [118], Hussain et al [63] and Siddiqui et al [123]. Herein, we do not explore these any further.…”

Section: Large Timementioning

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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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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“…Xu et al [36] applied the LTS-Godunov scheme to the 1D shallow water equations. Recently, Thompson and Moeller [34] independently rediscovered the LTS-Roe algorithm and applied it to Maxwell's equations.…”

Section: 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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A discontinuous wave-in-cell numerical scheme for hyperbolic conservation laws

Cited by 6 publications

References 29 publications

A Large Time Step Roe Scheme Applied to Two-Phase Flow

A Large Time Step Roe Scheme Applied to Two-Phase Flow

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

Large Time Step TVD Schemes for Hyperbolic Conservation Laws

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