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.
Accepté le :
DOI : 10.1051/m2an/2017051
Mots-clés : Large Time Step, HLL, HLLC, euler equations, riemann solver
@article{M2AN_2018__52_4_1239_0,
author = {Prebeg, Marin and Fl\r{a}tten, Tore and M\"uller, Bernhard},
title = {Large time step {HLL} and {HLLC} schemes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1239--1260},
publisher = {EDP-Sciences},
volume = {52},
number = {4},
year = {2018},
doi = {10.1051/m2an/2017051},
mrnumber = {3875285},
zbl = {1417.65160},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2017051/}
}
TY - JOUR AU - Prebeg, Marin AU - Flåtten, Tore AU - Müller, Bernhard TI - Large time step HLL and HLLC schemes JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1239 EP - 1260 VL - 52 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2017051/ DO - 10.1051/m2an/2017051 LA - en ID - M2AN_2018__52_4_1239_0 ER -
%0 Journal Article %A Prebeg, Marin %A Flåtten, Tore %A Müller, Bernhard %T Large time step HLL and HLLC schemes %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1239-1260 %V 52 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2017051/ %R 10.1051/m2an/2017051 %G en %F M2AN_2018__52_4_1239_0
Prebeg, Marin; Flåtten, Tore; Müller, Bernhard. Large time step HLL and HLLC schemes. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1239-1260. doi : 10.1051/m2an/2017051. http://www.numdam.org/articles/10.1051/m2an/2017051/
[1] , , and , On the choice of wavespeeds for the HLLC Riemann solver. SIAM J. Sci. Comput. 18 (1997) 1553–1570. | DOI | MR | Zbl
[2] and , On the computation of the Baer–Nunziato model using ALE formulation with HLL-and HLLC-type solvers towards fluid–structure interactions. J. Comput. Phys. 304 (2016) 189–230. | DOI | MR | Zbl
[3] , , and , Numerical experiments using a HLLC-type scheme with ALE formulation for compressible two-phase flows five-equation models with phase transition. Comput. Fluids 94 (2014) 112–138. | DOI | MR | Zbl
[4] , Simplified second-order Godunov-type methods. SIAM J. Sci. Stat. Comput. 9 (1988) 445–473. | DOI | MR | Zbl
[5] , On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25 (1988) 294–318. | DOI | MR | Zbl
[6] , and , On Godunov-type methods near low densities. J. Comput. Phys. 92 (1991) 273–295. | DOI | MR | Zbl
[7] , High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. | DOI | MR | Zbl
[8] , On a Large Time–Step high resolution scheme. Math. Comput. 46 (1986) 379–399. | DOI | MR | Zbl
[9] , and , On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. | DOI | MR | Zbl
[10] and , Conditions for the construction of multi-point total variationl diminishing difference schemes. Appl. Numer. Math. 2 (1986) 335–345. | DOI | MR | Zbl
[11] and , Corrigendum: “Conditions for the construction of multi-point total variation diminishing difference schemes”. Appl. Numer. Math. 3 (1987) 289. | DOI | MR | Zbl
[12] , A positive conservative method for magnetohydrodynamics based on HLL and Roe methods. J. Comput. Phys. 160 (2000) 649–661. | DOI | MR | Zbl
[13] , Large time step shock-capturing techniques for scalar conservation laws. SIAM J. Numer. Anal. 19 (1982) 1091–1109. | DOI | MR | Zbl
[14] , Convergence of a large time step generalization of Godunov’s method for conservation laws. Commut. Pure Appl. Math. 37 (1984) 463–477. | DOI | MR | Zbl
[15] , A large time step generalization of Godunov’s method for systems of conservation laws. SIAM J. Numer. Anal. 22 (1985) 1051–1073. | DOI | MR | Zbl
[16] , High resolution finite volume methods on arbitrary grids via wave propagation. J. Comput. Phys. 78 (1988) 36–63. | DOI | MR | Zbl
[17] , Finite Volume Methods for Hyperbolic Problems, 1st Edition, Book 31 of Cambridge Texts in Applied Mathematics. Cambridge University Press (2002). | MR | Zbl
[18] , , and , Large Time Step TVD schemes for hyperbolic conservation laws. SIAM J. Numer. Anal. 54 (2016) 2775–2798. | DOI | MR | Zbl
[19] and , A Large Time Step Roe scheme applied to two-phase flow. In VII European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS (Crete Island, Greece, 2016). Edited by , , and , (2016). | DOI
[20] , , and , HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer–Nunziato two-fluid model. J. Comput. Phys. 326 (2016) 733–762. | DOI | MR | Zbl
[21] and , Fast solution of time domain Maxwell’s equations using large time steps. In 2015 IEEE Inter. Confer. Comput. Electrom (ICCEM 2015) (Hong Kong, China, 2015), Institute of Electrical and Electronics Engineers (2015) 330–332.
[22] and , A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208 (2005) 315–344. | DOI | MR | Zbl
[23] , and , A large time step 1D upwind explicit scheme (CFL>1): Application to shallow water equations. J. Comput. Phys. 231 (2012) 6532–6557. | DOI | MR | Zbl
[24] , and , A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries. J. Comput. Phys. 263 (2014) 303–327. | DOI | MR | Zbl
[25] , , and , A Large Time Step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws: Application to the homogeneous shallow water equations. Appl. Math. Model. 47 (2017) 294–317. | DOI | MR | Zbl
[26] , , and , A large time step upwind scheme for the shallow water equations with source terms. In Numerical Methods for Hyperbolic Equations, edited by , , and . CRC Press, (2012) 141–148. | DOI | MR
[27] , , and , Extension of an explicit finite volume method to large time steps (CFL>1): Application to shallow water flows. Int. J. Numer. Meth. Fluids 50 (2006) 63–102. | DOI | MR | Zbl
[28] and , A mixture-energy-consistent six-equation two-phase numerical model for fluids with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259 (2014) 331–357. | DOI | MR | Zbl
[29] , Numerical viscosity in Large Time Step HLL-type schemes. In Proc. of the Sixteenth International Conference on Hyperbolic Problems, HYP2016 (Aachen, Germany, 2017), edited by and . Springer (2018) 479–490. | MR | Zbl
[30] , and , Large Time Step Roe scheme for a common 1D two-fluid model. Appl. Math. Model. 44 (2017) 124–142. | DOI | MR | Zbl
[31] and , A class of large time step Godunov schemes for hyperbolic conservation laws and applications. J. Comput. Phys. 230 (2011) 7418–7440. | DOI | MR | Zbl
[32] , Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357–372. | DOI | MR | Zbl
[33] , A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27 (1978) 1–31. | DOI | MR | Zbl
[34] , Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43 (1984) 369–381. | DOI | MR | Zbl
[35] , and , Combining Discrete Equations Method and upwind downwind-controlled splitting for non-reacting and reacting two-fluid computations: One dimensional case. Comput. Fluids 93 (2014) 74–90. | DOI | MR | Zbl
[36] , , and , A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver. Comput. Fluids 46 (2011) 122–132. | DOI | MR | Zbl
[37] and , HLLC-type Riemann solver for Baer–Nunziato equations of compressible two-phase flow. J. Comput. Phys. 229 (2010) 3573–3604. | DOI | MR | Zbl
[38] , Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition. Springer–Verlag Berlin Heidelberg (2009). | DOI | MR
[39] , and , Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4 (1994) 25–34. | DOI | Zbl
[40] and , The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1984) 115–173. | DOI | MR | Zbl
[41] , , , and , A large time step Godunov scheme for free-surface shallow water equations. Chinese Sci. Bull. 59 (2014) 2534–2540. | DOI
[42] and , Two-dimensional two-fluid two-phase flow simulation using an approximate Jacobian matrix for HLL scheme. Numer. Heat Tran. B 56 (2010) 72–392.
Cité par Sources :
