J. Trangenstein, *Numerical Solution of Hyperbolic Partial Differential Equations*, Cambridge University Press, Cambridge, 2008.Google Scholar

J. von Neumann and R.D. Richtmeyer, "A method for the numerical calculation of hydrodynamic shocks", *J. Appl. Phys.* 21, 232-237 (1950).Google Scholar

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).Google Scholar

K. Murawski and R.S. Steinolfson, "Numerical modeling of the solar wind interaction with Venus", *Planet. Space Sci.* 44 (3), 243-252 (1996).Google Scholar

S.K. Godunov, "A difference scheme for numerical solution of discontinuos solution of hydrodynamic equations", *Math. Sb.* 47, 271-306 (1959).Google Scholar

D. Lee and A.E. Deane, "An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics", *J. Comput. Phys.* 228 (4), 952-975 (2009).Web of ScienceGoogle Scholar

J.J. Quirk, "An adaptive grid algorithm for computational shock hydrodynamics", *PhD Thesis*, College of Aeronautics, Cranfield Institute of Technology, Cranfield, 1991.Google Scholar

E. Toro, *Riemann Solvers and Numerical Methods for Fluid Dynamics*, Springer, Berlin, 2009.Google Scholar

R.J. LeVeque, *Numerical Methods for Conservation Laws*, Birkhäuser Verlag Basel, Berlin, 1990.Google Scholar

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).CrossrefGoogle Scholar

B. Einfeld, "On Godunov-type methods for gas dynamics", *SIAM J. Num. Anal.* 25 (2), 294-318 (1988).Google Scholar

P.L. Roe, "Approximate Riemann solvers, parameter vectors and difference schemes", *J. Comp. Phys.* 43, 357-372 (1981).Google Scholar

N. Aslan, "Two-dimensional solutions of MHD equations with an adapted Roe method", *Int. J. Numer. Meth. Fluids* 23 (11), 1211-1222 (1996).Google Scholar

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).Google Scholar

R. Donat and A. Marquina, "Capturing shock reflections: an improved flux formula", *J. Comp. Phys.* 125 (1), 42-58 (1996).Google Scholar

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).Google Scholar

R.J. LeVeque, *Finite-volume Methods for Hyperbolic Problems*, Cambridge University Press, Cambridge, 2002.Google Scholar

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).Google Scholar

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).Google Scholar

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).Google Scholar

P.L. Roe, "Sonic flux formulae", *SIAM J. Sci. Stat. Comput.* 13, 611-630 (1982).Google Scholar

A. Harten, "ENO schemes with subcell resolution", *J. Comp. Phys.* 83, 148-184 (1989).Google Scholar

P.R. Woodward and P. Colella, "The numerical simulation of two-dimensional fluid flow with strong shocks", *J. Comp. Phys.* 54, 115-173 (1984).Google Scholar

K. Murawski and M. Goossens, "Operator splitting for multidimensional magnetohydrodynamics", *J. Geophys. Res.* 99 (A6), 11569-11573 (1994).Google Scholar

G. Strang, "On the construction and comparison of difference schemes", *SIAM J. Num. Anal.* 5 (3), 506-517 (1968).Google Scholar

P. Colella, "Multidimensional upwind methods for hyperbolic conservation laws", *J. Comp. Phys.* 87, 171-200 (1990).Google Scholar

M.J. Berger and P. Colella, "Local adaptive mesh refinement for shock hydrodynamics", *J. Comp. Phys.* 82, 64-84 (1989).Google Scholar

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).Google Scholar

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).Google Scholar

D. DeZeeuw and K.G. Powell, "An adaptively refined Cartesian mesh solver for the Euler equations", *J. Comp. Phys.* 104, 56-68 (1993).Google Scholar

J.J. Quirk, "A cartesian grid approach with hierarchical refinement for compressible flows", *Computers and Fluids* 23 (1), 125-142 (1994).Google Scholar

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).Google Scholar

S. Arendt, "Vorticity in stratified fluids. I. General formulation", *Geophys. Astrophys. Fluid Dynamics* 68 (1), 59-83 (1993).Google Scholar

J.V. Hollweg, "On the origin of solar spicules", *Astrophys. J.* 257, 345-353 (1982).Google Scholar

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