## Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows.(English)Zbl 1303.76140

Summary: We present a general strategy for constructing multidimensional HLLE Riemann solvers, with particular attention paid to detailing the two-dimensional HLLE Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow.
We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical (MHD) flows. A robust and efficient second order accurate numerical scheme for two and three-dimensional Euler and MHD flows is presented. The scheme is built on the current multidimensional Riemann solver and has been implemented in the author’s RIEMANN code. The number of zones updated per second by this scheme on a modern processor is shown to be cost-competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps.
Accuracy analysis for multidimensional Euler and MHD problems shows that the scheme meets its design accuracy. Several stringent test problems involving Euler and MHD flows are also presented and the scheme is shown to perform robustly on all of them.

### MSC:

 76W05 Magnetohydrodynamics and electrohydrodynamics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

RIEMANN; HLLE
Full Text:

### References:

 [1] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des èquations d’euler par une methode de type roe, I: la linèarisation, C.R. acad. sci. ser. I, 319, 499, (1994) · Zbl 0813.76074 [2] Abgrall, R., Approximation du problème de Riemann vraiment multidimensionnel des èquations d’euler par une methode de type roe, II: solution du probleme de Riemann approchè, C.R. acad. sci. ser. I, 319, 625, (1994) · Zbl 0813.76075 [3] Balsara, D.S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. suppl., 116, 119, (1998) [4] Balsara, D.S., Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. comput. phys., 174, 614-648, (2001) · Zbl 1157.76369 [5] Balsara, D.S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. suppl., 151, 149-184, (2004) [6] Balsara, D.S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. comput. phys., 228, 5040-5056, (2009) · Zbl 1280.76030 [7] Balsara, D.S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. comput. phys., 228, 2480-2516, (2009) · Zbl 1275.76169 [8] Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys., 160, 405-452, (2000) · Zbl 0961.65078 [9] Balsara, D.S.; Spicer, D.S., A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. comput. phys., 149, 270-292, (1999) · Zbl 0936.76051 [10] Batten, P.; Clarke, N.; Lambert, C.; Causon, D.M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. sci. comput., 18, 1553-1570, (1997) · Zbl 0992.65088 [11] Brackbill, J.U.; Barnes, D.C., The effect of nonzero $$\mathbf{\nabla} \cdot \boldsymbol{B}$$ on the numerical solution of the magnetohydrodynamic equations, J. comput. phys., 35, 426-430, (1980) · Zbl 0429.76079 [12] Brecht, S.H.; Lyon, J.G.; Fedder, J.A.; Hain, K., A simulation study of east-west IMF effects on the magnetosphere, Geophys. res. lett., 8, 397, (1981) [13] Brio, M.; Zakharian, A.R.; Webb, G.M., Two-dimensional Riemann solver for Euler equations of gas dynamics, J. comput. phys., 167, 177-195, (2001) · Zbl 1043.76042 [14] Cargo, P.; Gallice, G., Roe matrices for ideal MHD and systematic construction of roe matrices for systems of conservation laws, J. comput. phys., 136, 446, (1997) · Zbl 0919.76053 [15] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. sci. statist. comput., 6, 104, (1985) · Zbl 0562.76072 [16] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. comput. phys., 87, 171, (1990) · Zbl 0694.65041 [17] Colella, P.; Woodward, P.R., The piecewise parabolic method (PPM) for gas-dynamical simulations, J. comput. phys., 54, 174-201, (1984) · Zbl 0531.76082 [18] Dai, W.; Woodward, P.R., On the divergence-free condition and conservation laws in numerical simulations for supersonic magnetohydrodynamic flows, Astrophys. J., 494, 317-335, (1998) [19] Dedner, A.; Kemm, F.; Kröener, D.; Munz, C.-D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for MHD equations, J. comput. phys., 175, 645-673, (2002) · Zbl 1059.76040 [20] DeVore, C.R., Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, J. comput. phys., 92, 142-160, (1991) · Zbl 0716.76056 [21] Einfeldt, B., On Godunov-type methods for gas dynamics, SIAM J. numer. anal., 25, 3, 294-318, (1988) · Zbl 0642.76088 [22] Einfeldt, B.; Munz, C.-D.; Roe, P.L.; Sjogreen, B., On Godunov-type methods near low densities, J. comput. phys., 92, 273-295, (1991) · Zbl 0709.76102 [23] Evans, C.R.; Hawley, J.F., Simulation of magnetohydrodynamic flows: a constrained transport method, Astrophys. J., 332, 659, (1989) [24] Fey, M., Multidimensional upwinding 1. the method of transport for solving the Euler equations, J. comput. phys., 143, 159, (1998) · Zbl 0932.76050 [25] Fey, M., Multidimensional upwinding 2. decomposition of the Euler equation into advection equation, J. comput. phys., 143, 181, (1998) · Zbl 0932.76051 [26] Fuchs, F.; Mishra, S.; Risebro, N.H., Splitting based finite volume schemes for the ideal MHD equations, J. comput. phys., 228, 3, 641-660, (2009) · Zbl 1259.76021 [27] Gardiner, T.; Stone, J.M., An unsplit Godunov method for ideal MHD via constrained transport, J. comput. phys., 295, 509, (2005) · Zbl 1087.76536 [28] Gilquin, H.; Laurens, J.; Rosier, C., Multidimensional Riemann problems for linear hyperbolic systems, Notes numer. fluid mech., 43, 284, (1993) · Zbl 0921.35090 [29] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. USSR, sbornik., 47, 271-306, (1959) · Zbl 0171.46204 [30] Gurski, K.F., An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics, SIAM J. sci. comput., 25, 2165, (2004) · Zbl 1133.76358 [31] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 289-315, (1983) · Zbl 0565.65051 [32] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067 [33] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065 [34] Kurganov, A.; Noelle, S.; Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and hamilton – jacobi equations, SIAM J. sci. comput., 23, 4, 707-740, (2001) · Zbl 0998.65091 [35] LeVeque, R.J., Wave propagation algorithms for multidimensional hyperbolic systems, J. comput. phys., 131, 327, (1997) · Zbl 0872.76075 [36] Londrillo, P.; DelZanna, L., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. comput. phys., 195, 17-48, (2004) · Zbl 1087.76074 [37] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. comput. phys., 208, 315-344, (2005) · Zbl 1114.76378 [38] Orszag, S.A.; Tang, C.M., Small-scale structure of two-dimensional magnetohydrodynamic turbulence, J. fluid mech., 90, 129, (1979) [39] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. comput., 38, 158, 339, (1982) · Zbl 0483.65055 [40] K.G. Powell, An approximate Riemann solver for MHD ( that actually works in more than one dimension), ICASE Report No. 94-24, Langley VA, 1994. [41] Roe, P.L., Approximate Riemann solver, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066 [42] Roe, P.L., Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. comput. phys., 63, 458, (1986) · Zbl 0587.76126 [43] Roe, P.L.; Balsara, D.S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. appl. math., 56, 57, (1996) · Zbl 0845.35092 [44] Roe, P.L.; Sidilkover, D., Optimum positive linear schemes for advection in two and three dimensions, SIAM J. numer. anal., 29, 1542-1568, (1992) · Zbl 0765.65093 [45] Rumsey, C.B.; van Leer, B.; Roe, P.L., A multidimensional flux function with application to the Euler and navier – stokes equations, J. comput. phys., 105, 306, (1993) · Zbl 0767.76039 [46] Rusanov, V.V., Calculation of interaction of non-steady shock waves with obstacles, J. comput. math. phys. USSR, 1, 267, (1961) [47] Ryu, D.; Miniati, F.; Jones, T.W.; Frank, A., A divergence-free upwind code for multidimensional magnetohydrodynamic flows, Astrophys. J., 509, 244-255, (1998) [48] Saltzman, J., An unsplit 3D upwind method for hyperbolic conservation laws, J. comput. phys., 115, 153, (1994) · Zbl 0813.65111 [49] Schulz-Rinne, C.W.; Collins, J.P.; Glaz, H.M., Numerical solution of the Riemann problem for two-dimensional gas dynamics, SIAM J. sci. comput., 14, 7, 1394-1414, (1993) · Zbl 0785.76050 [50] Toro, E.F.; Spruce, M.; Speares, W., Restoration of contact surface in the HLL Riemann solver, Shock waves, 4, 25-34, (1994) · Zbl 0811.76053 [51] van Leer, B., Toward the ultimate conservative difference scheme: V. A second-order sequel to godunov’s method, J. comput. phys., 32, 101, (1979) · Zbl 1364.65223 [52] Wendroff, B., A two-dimensional HLLE Riemann solver and associated Godunov-type difference scheme for gas dynamics, Comput. math. appl., 38, 175-185, (1999) · Zbl 0984.76064 [53] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
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