We present Large Time Step (LTS) extensions of the Harten-Lax-van Leer (HLL) and Harten-Lax-van Leer-Contact (HLLC) schemes. Herein, LTS denotes a class of explicit methods stable for Courant numbers greater than one. The original LTS method (R.J. LeVeque, SIAM J. Numer. Anal. 22 (1985) 1051–1073) was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe's approximate Riemann solver. In this paper, we formulate the LTS extension of the HLL and HLLC schemes in conservation form. We provide explicit expressions for the flux-difference splitting coefficients and the numerical viscosity coefficients of the LTS-HLL scheme. We apply the new schemes to the one-dimensional Euler equations and compare them to their non-LTS counterparts. As test cases, we consider the classical Sod shock tube problem and the Woodward-Colella blast-wave problem. We numerically demonstrate that for the right choice of wave velocity estimates both schemes calculate entropy satisfying solutions.
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
We consider Large Time Step (LTS) methods, i.e. the explicit finite volume methods not limited by the CFL (Courant-Friedrichs-Lewy) condition. The original LTS method [R. J. LeVeque, SIAM J. Numer. Anal., (22) 1985] was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe's approximate Riemann solver. Recently, Prebeg et al. [in press, 2017] developed the LTS extension of the HLL and HLLC schemes. We perform the modified equation analysis and demonstrate that for the appropriate choice of the wave velocity estimates the LTS-HLL scheme yields entropy satisfying solutions. We apply the LTS-HLL(C) schemes to the one-dimensional Euler equations and consider the Sod shock tube, double rarefaction and Woodward-Colella blast-wave problem.
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