zbMATH — the first resource for mathematics

An upwind CESE scheme for 2D and 3D MHD numerical simulation in general curvilinear coordinates. (English) Zbl 1415.76509
Summary: In [ibid. 288, 101–118 (2015; Zbl 1354.65197); ibid. 305, 775–792 (2016; Zbl 1349.65483)], H. Shen et al. proposed an upwind space-time conservation element and solution element (CESE) scheme for 1D and 2D hydrodynamics (HD) in rectangular coordinates, which combined the advantages of CESE and upwind scheme, namely, guaranteed strictly the space-time conservation law as well as captured discontinuities very efficiently. All kinds of upwind schemes can be combined very flexibly for different problems to achieve the perfect combination of CESE and finite volume method (FVM). However, in many physical applications, we need to consider geometries that are more sophisticated. Hence, the main objective of this paper is to extend the upwind CESE scheme to multidimensional magneto-hydrodynamics (MHD) in general curvilinear coordinates by transforming the MHD equations from the physical domain (general curvilinear coordinates) to the computational domain (rectangular coordinates) and the new equations in the computational domain can be still written in the conservation form. For the 3D case, the derivations of some formulas are much more abstract and complex in a 4D Euclidean hyperspace, and some technical problems need to be solved in the debugging process. Unlike in HD, keeping the magnetic field divergence-free for MHD problems is also a challenge especially in general curvilinear coordinates. These are the main obstacles we have overcome in this study. The test results of benchmarks demonstrate that we have successfully extended the upwind CESE scheme to general curvilinear coordinates for both 2D and 3D MHD problems.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76W05 Magnetohydrodynamics and electrohydrodynamics
76L05 Shock waves and blast waves in fluid mechanics
Full Text: DOI
[1] Shen, H.; Wen, C. Y.; Zhang, D. L., A characteristic space-time conservation element and solution element method for conservation laws, J. Comput. Phys., 288, 101-118, (2015) · Zbl 1354.65197
[2] Shen, H.; Wen, C. Y., A characteristic space-time conservation element and solution element method for conservation laws II, J. Comput. Phys., 305, 775-792, (2016) · Zbl 1349.65483
[3] Evje, S.; Flatten, T., Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192, 175-210, (2003) · Zbl 1032.76696
[4] , , www.cfluid.com or http://cid-1cc0dcbff560c149.skydrive.live.com/browse.aspx/.Public.
[5] Liou, M. S.; Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107, 23-39, (1993) · Zbl 0779.76056
[6] Liou, M. S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 364-382, (1996) · Zbl 0870.76049
[7] Jameson, A., Analysis and design of numerical schemes for gas dynamics, I: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flow, J. Comput. Fluid Dyn., 4, 171-218, (1995)
[8] Jameson, A., Analysis and design of numerical schemes for gas dynamics, II: artificial diffusion and discrete shock structure, J. Comput. Fluid Dyn., 5, 1-38, (1995)
[9] Zha, G. C.; Bilgen, E., Numerical solutions of Euler equations by using a new flux vector splitting scheme, Int. J. Numer. Methods Fluids, 17, 115-144, (1993) · Zbl 0779.76067
[10] Zha, G. C.; Shen, Y. Q.; Wang, B. Y., An improved low diffusion E-CUSP upwind scheme, Comput. Fluids, 48, 214-220, (2011) · Zbl 1271.76200
[11] Shen, Y. Q.; Zha, G. C.; Huerta, M. A., E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO scheme, J. Comput. Phys., 231, 6233-6247, (2012)
[12] Edwards, J. R., A low-diffusion flux-splitting scheme for Navier-Stokes calculations, Comput. Fluids, 6, 635-659, (1997) · Zbl 0911.76055
[13] Sun, M.; Takayama, K., An artificially upstream flux vector splitting scheme for the Euler equations, J. Comput. Phys., 189, 305-329, (2003) · Zbl 1097.76574
[14] Steger, J. L.; Warming, R. F., Flux vector splitting of the inviscid gasdynamic equations with application to finite difference method, J. Comput. Phys., 40, 263-293, (1981) · Zbl 0468.76066
[15] Li, B.; Yuan, L., Convergence issue in using high-resolution schemes and lower-upper symmetric Gauss-Seidel method for steady shock-induced combustion problems, Int. J. Numer. Methods Fluids, 71, 1422-1437, (2013)
[16] Liou, M. S.; van Leer, B.; Shuen, J. S., Splitting of inviscid fluxes for real gases, J. Comput. Phys., 87, 1-24, (1990) · Zbl 0687.76074
[17] MacCormack, R. W., An upwind conservation form method for ideal magnetohydrodynamics equations, (1999), AIAA, Paper 99-3609
[18] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61, (1983) · Zbl 0565.65051
[19] Balsara, D. S., Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamical flows, J. Comput. Phys., 229, 1970-1993, (2010) · Zbl 1303.76140
[20] Balsara, D. S.; Dumbser, M.; Abgrall, R., Multidimensional HLL and HLLC Riemann solvers for unstructured meshes - with application to Euler and MHD flows, J. Comput. Phys., 261, 172-208, (2014) · Zbl 1349.76426
[21] Janhunen, P., A positive conservative method for magnetohydrodynamics based on HLL and roe methods, J. Comput. Phys., 160, 649-661, (2000) · Zbl 0967.76061
[22] Toro, E. F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL Riemann solver, Shock Waves, 4, 25-34, (1994) · Zbl 0811.76053
[23] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (2009), Springer · Zbl 1227.76006
[24] Gurski, K. F., An HLLC-type approximate Riemann solver for ideal magnetohydrodynamics, SIAM J. Sci. Comput., 25, 2165-2187, (2004) · Zbl 1133.76358
[25] Li, S. T., An HLLC Riemann solver for magnetohydrodynamics, J. Comput. Phys., 203, 344-357, (2005) · Zbl 1299.76302
[26] Miyoshi, T.; Kusano, K., A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics, J. Comput. Phys., 208, 315-344, (2005) · Zbl 1114.76378
[27] Mignone, A.; Ugliano, M.; Bodo, G., A five-wave harten-Lax-Van leer Riemann solver for relativistic magnetohydrodynamics, Mon. Not. R. Astron. Soc., 393, 1141-1156, (2009)
[28] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[29] Cargo, P.; Gallice, G., Roe matrices for ideal MHD and systematic construction of roe matrices for systems of conservation laws, J. Comput. Phys., 136, 446-466, (1997) · Zbl 0919.76053
[30] Balsara, D. S., Total variation diminishing scheme for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 133-153, (1998)
[31] Roe, P. L.; Balsara, D. S., Notes on the eigensystem of magnetohydrodynamics, SIAM J. Appl. Math., 56, 57-67, (1996) · Zbl 0845.35092
[32] Balsara, D. S., Linearized formulation of the Riemann problem for adiabatic and isothermal magnetohydrodynamics, Astrophys. J. Suppl. Ser., 116, 119-131, (1998)
[33] Tchuen, G.; Fogang, F.; Burtschell, Y.; Woafo, P., A hybrid numerical method and its application to inviscid compressible flow problems, Comput. Phys. Commun., 185, 479-488, (2014) · Zbl 1348.76126
[34] Fogang, F.; Tchuen, G.; Burtschell, Y.; Woafo, P., An extension of AUFSR scheme for the ideal magnetohydrodynamics equations, Comput. Fluids, 114, 297-313, (2015) · Zbl 1390.76915
[35] Rossow, C. C., A flux-splitting scheme for compressible and incompressible flows, J. Comput. Phys., 164, 104-122, (2000) · Zbl 0991.76048
[36] Coquel, F.; Liou, M. S., Hybrid upwind splitting (HUS) by a field by field decomposition, (1995), NASA, ICOMP-95-2
[37] Kitamura, K.; Nonomura, T., Simple and robust HLLC extensions of two-fluid AUSM for multiphase flow computations, Comput. Fluids, 100, 321-335, (2014) · Zbl 1391.76480
[38] Kapen, P. T.; Ghislain, T., A new flux splitting scheme based on toro-vazquez and HLL scheme for the Euler equations, J. Comput. Methods Phys., 2014, (2014) · Zbl 1318.76014
[39] Xisto, C. M.; Ṕascoa, J. C.; Oliveira, P. J., A pressure-based method with AUSM-type fluxes for MHD flows at arbitrary Mach numbers, Int. J. Numer. Methods Fluids, 72, 1165-1182, (2013)
[40] Xisto, C. M.; Ṕascoa, J. C.; Oliveira, P. J.; Nicolini, D. A., A hybrid pressure-density-based algorithm for the Euler equations at all Mach number regimes, Int. J. Numer. Methods Fluids, 70, 961-976, (2012)
[41] Nishikawa, H.; Kitamura, K., Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers, J. Comput. Phys., 227, 2560-2581, (2008) · Zbl 1388.76185
[42] Ren, Y. X., A robust shock-capturing scheme based on rotated Riemann solvers, Comput. Fluids, 32, 1379-1403, (2003) · Zbl 1034.76035
[43] Levy, D. W.; Powell, K. G.; Leer, B. V., Use of a rotated solver for the two-dimensional Euler equations, J. Comput. Phys., 106, 201-214, (1993) · Zbl 0770.76046
[44] Chang, S. C., The method of space-time conservation element and solution element - a new approach for solving the Navier-Stokes and Euler equations, J. Comput. Phys., 119, 295-324, (1995) · Zbl 0847.76062
[45] Wang, X. Y., A summary of the space-time conservation element and solution element (CESE), method, (2015), NASA/TM-2015-218743
[46] Huynh, H. T., Analysis and improvement of upwind and centered schemes on quadrilateral and triangular meshes, (2003), AIAA 2003-3541
[47] Vinokur, M., Conservation equations of gas-dynamics in curvilinear coordinate systems, J. Comput. Phys., 14, 105-125, (1974) · Zbl 0277.76061
[48] Viviand, H., Conservation forms of gas dynamic equations, 153-159, (1974), European Space Research Organization Technical Translation, ESRO-TT-144, 1
[49] Bilyeu, D., A high-order conservation element solution element method for solving hyperbolic differential equations on unstructured grid, (2014), University of the Ohio State, PhD Thesis
[50] Zhang, Z. C.; John Yu, S. T.; Chang, S. C., A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. Comput. Phys., 175, 168-199, (2002) · Zbl 1168.76339
[51] Feng, X. S.; Hu, Y. Q.; Wei, F. S., Modeling the resistive MHD by the CESE method, Sol. Phys., 235, 235-257, (2006)
[52] Yang, Y.; Feng, X. S.; Jiang, C. W., A high-order CESE scheme with a new divergence-free method for MHD numerical simulation, J. Comput. Phys., 349, 561-581, (2017) · Zbl 1380.76112
[53] Balsara, D. S.; Dumbser, M., Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715, (2015) · Zbl 1351.76092
[54] Skinner, M. A.; Ostriker, E. C., The athena astrophysical magnetohydrodynamics code in cylindrical geometry, Astrophys. J. Suppl. Ser., 188, 290-311, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.