Numerical methods of solving equations of hydrodynamics from perspectives of the code FLASH. (English) Zbl 1291.76219
Summary: In this paper we review numerical methods for hydrodynamic equations. Internal complexity make numerical solutions of these equations a formidable task. We present results of advanced numerical simulations for a complex system with a use of a publicly available code, FLASH. These results proof that the numerical methods cope very well with this task.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
References:
| [1] | J. Trangenstein, Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge University Press, Cambridge, 2008. |
| [2] | J. von Neumann and R.D. Richtmeyer, ”A method for the numerical calculation of hydrodynamic shocks”, J. Appl. Phys. 21, 232-237 (1950). · Zbl 0037.12002 · doi:10.1063/1.1699639 |
| [3] | J.M. Stone and M.L. Norman, ”ZEUS-2D: a radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II. The magnetohydrodynamic algorithms and tests”, Astrophys. J. Suppl. Ser. 80, 791-818 (1992). |
| [4] | K. Murawski and R.S. Steinolfson, ”Numerical modeling of the solar wind interaction with Venus”, Planet. Space Sci. 44 (3), 243-252 (1996). |
| [5] | S.K. Godunov, ”A difference scheme for numerical solution of discontinuos solution of hydrodynamic equations”, Math. Sb. 47, 271-306 (1959). · Zbl 0171.46204 |
| [6] | D. Lee and A.E. Deane, ”An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics”, J. Comput. Phys. 228 (4), 952-975 (2009). · Zbl 1330.76093 · doi:10.1016/j.jcp.2008.08.026 |
| [7] | J.J. Quirk, ”An adaptive grid algorithm for computational shock hydrodynamics”, PhD Thesis, College of Aeronautics, Cranfield Institute of Technology, Cranfield, 1991. |
| [8] | E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin, 2009. · Zbl 1227.76006 · doi:10.1007/b79761 |
| [9] | R.J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag Basel, Berlin, 1990. · Zbl 0723.65067 |
| [10] | A. Harten, P.D. Lax, and B. van Leer, ”On upstream differencing and Godunov-type schemes for hyperbolic conservation laws”, SIAM Rev. 25 (1), 35-61 (1983). · Zbl 0565.65051 · doi:10.1137/1025002 |
| [11] | B. Einfeld, ”On Godunov-type methods for gas dynamics”, SIAM J. Num. Anal. 25 (2), 294-318 (1988). |
| [12] | P.L. Roe, ”Approximate Riemann solvers, parameter vectors and difference schemes”, J. Comp. Phys. 43, 357-372 (1981). · Zbl 0474.65066 · doi:10.1016/0021-9991(81)90128-5 |
| [13] | N. Aslan, ”Two-dimensional solutions of MHD equations with an adapted Roe method”, Int. J. Numer. Meth. Fluids 23 (11), 1211-1222 (1996). · Zbl 0882.76052 · doi:10.1002/(SICI)1097-0363(19961215)23:11<1211::AID-FLD469>3.0.CO;2-5 |
| [14] | B. Einfeld, C.D. Munz, P.L. Roe, amd B. Sjögreen, ”On Godunov-type methods near low densities”, J. Comp. Phys. 92 (2), 273-295 (1991). |
| [15] | R. Donat and A. Marquina, ”Capturing shock reflections: an improved flux formula”, J. Comp. Phys. 125 (1), 42-58 (1996). · Zbl 0847.76049 · doi:10.1006/jcph.1996.0078 |
| [16] | S. Jin and Z.P. Xin, ”The relaxation schemes for systems of conservation laws in arbitrary space dimensions”, Comm. Pure Appl. Math. 48 (3), 235-276 (1995). · Zbl 0826.65078 · doi:10.1002/cpa.3160480303 |
| [17] | R.J. LeVeque, Finite-volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. · Zbl 1010.65040 · doi:10.1017/CBO9780511791253 |
| [18] | R.J. LeVeque and M. Pelanti, ”A class of approximate Riemann solvers and their relation to relaxation schemes”, J. Comput. Phys. 172 (2), 572-591 (2001). · Zbl 0988.65072 · doi:10.1006/jcph.2001.6838 |
| [19] | V.P. Kolgan, ”Application of the minimum-derivative principle in the construction of finite-diverence schemes for numerical analysis of discontinuous solutions in gas dynamics”, Uch. Zap. TsaGI 3 (6), 68-77 (1972). |
| [20] | B. van Leer, ”Towards the ultimate conservative difference scheme. V-A second-order sequel to Godunov’s method”, J. Comp. Phys. 32, 101-136 (1979). · Zbl 1364.65223 |
| [21] | P.L. Roe, ”Sonic flux formulae”, SIAM J. Sci. Stat. Comput. 13, 611-630 (1982). · Zbl 0747.65073 · doi:10.1137/0913034 |
| [22] | A. Harten, ”ENO schemes with subcell resolution”, J. Comp. Phys. 83, 148-184 (1989). · Zbl 0696.65078 · doi:10.1016/0021-9991(89)90226-X |
| [23] | P.R. Woodward and P. Colella, ”The numerical simulation of two-dimensional fluid flow with strong shocks”, J. Comp. Phys. 54, 115-173 (1984). · Zbl 0573.76057 · doi:10.1016/0021-9991(84)90142-6 |
| [24] | K. Murawski and M. Goossens, ”Operator splitting for multidimensional magnetohydrodynamics”, J. Geophys. Res. 99 (A6), 11569-11573 (1994). |
| [25] | G. Strang, ”On the construction and comparison of difference schemes”, SIAM J. Num. Anal. 5 (3), 506-517 (1968). · Zbl 0184.38503 · doi:10.1137/0705041 |
| [26] | P. Colella, ”Multidimensional upwind methods for hyperbolic conservation laws”, J. Comp. Phys. 87, 171-200 (1990). · Zbl 0694.65041 · doi:10.1016/0021-9991(90)90233-Q |
| [27] | M.J. Berger and P. Colella, ”Local adaptive mesh refinement for shock hydrodynamics”, J. Comp. Phys. 82, 64-84 (1989). · Zbl 0665.76070 · doi:10.1016/0021-9991(89)90035-1 |
| [28] | J. Bell, M. Berger, J. Saltzman, and M. Welcome, ”Three dimensional adaptive mesh refinement for hyperbolic conservation laws”, SIAM J. Sci. Comput. 15 (1), 127-138 (1994). · Zbl 0793.65072 · doi:10.1137/0915008 |
| [29] | D.F. Martin and P. Colella, ”A cell-centered adaptive projection method for the incompressible Euler equations”, J. Comp. Phys. 163 (2), 271-312 (2000). · Zbl 0991.76052 · doi:10.1006/jcph.2000.6575 |
| [30] | D. DeZeeuw and K.G. Powell, ”An adaptively refined Cartesian mesh solver for the Euler equations”, J. Comp. Phys. 104, 56-68 (1993). · Zbl 0766.76066 · doi:10.1006/jcph.1993.1007 |
| [31] | J.J. Quirk, ”A cartesian grid approach with hierarchical refinement for compressible flows”, Computers and Fluids 23 (1), 125-142 (1994). · Zbl 0788.76067 |
| [32] | J.-Y. Trepanier, M. Reggio, and D. Ait-Ali-Yahia, ”An implicit flux- difference splitting method for solving the Euler equations on adaptive traingular grids”, Int. J. Num. Meth. Heat Fluid Flow 3 (1), 63-77 (1993). |
| [33] | S. Arendt, ”Vorticity in stratified fluids. I. General formulation”, Geophys. Astrophys. Fluid Dynamics 68 (1), 59-83 (1993). |
| [34] | J.V. Hollweg, ”On the origin of solar spicules”, Astrophys. J. 257, 345-353 (1982). |
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