A class of large time step Godunov schemes for hyperbolic conservation laws and applications

Qian, Zhen; Lee, Chun‐Hian

doi:10.1016/j.jcp.2011.06.008

Cited by 34 publications

(32 citation statements)

References 16 publications

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“…In 2014, Xu et al [148] applied the LTS-Godunov scheme to the shallow water equations. Further applications of the LTS-Godunov scheme in recent years include the three-dimensional Euler equations by Qian and Lee [117], high speed combustion waves by Tang et al [134], and Maxwell's equations by Makwana and Chatterjee [95], while Lindqvist and Lund [90] applied the LTS-Roe scheme to two-phase flow model.…”

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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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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“…The wave interactions are assumed to be linear, which removes the need to handle wave interactions explicitly. More recently, Qian and Lee [21] extended this scheme by taking wave interactions into account. Norman et al [20] developed a LTS scheme similar to LeVeque's approach, based on a WENO discretization.…”

Section: Introductionmentioning

confidence: 99%

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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“…Once the number of boundary cells is determined, the information to be updated at each cell is obtained from the extrapolation of the boundary information through the characteristic lines. This treatment can be understood as the imposition of ghost cells, considered in [17] or [1] in the context of LTS schemes.…”

Section: Boundary Conditions For the Scalar Casementioning

confidence: 99%

“…Explicit Large Time Step (LTS) schemes are being increasingly used in the context of Computational Fluid Dynamics [1,2,3]. Apart from retaining most of the advantages offered by explicit schemes, they are able to increase not only the efficiency in terms of computational burden, but also the accuracy of the numerical results.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the extension of the LTS scheme to the 2D shallow water equations was firstly mentioned in [10]. Other recent applications in connection with atmospheric dynamics [2] and Euler equations [1] have been extended to more than one dimension using the dimensional splitting technique. In this work, the extension of the mentioned LTS scheme to the 2D shallow water equations is achieved by means of this dimensional splitting procedure on structured grids.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

Morales-Hernändez

Hubbard

García‐Navarro

2014

Journal of Computational Physics

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A 2D Large TimeStep (LTS) explicit scheme on structured grids is presented in this work. It is first detailed and analysed for the 2D linear advection equation and then applied to the 2D shallow water equations. The dimensional splitting technique allows us to extend the ideas developed in the 1D case related to source terms, boundary conditions and the reduction of the time step in the presence of large discontinuities. The boundary conditions treatment as well as the wet/dry fronts in the case of the 2D shallow water equations require extra effort. The proposed scheme is tested on linear and non-linear equations and systems, with and without source terms. The numerical results are compared with those of the conventional scheme as well as with analytical solutions and experimental data.

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A class of large time step Godunov schemes for hyperbolic conservation laws and applications

Cited by 34 publications

References 16 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

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

A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

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