Explicit-implicit schemes for the numerical solution of nonlinear hyperbolic systems

McGuire, G. R.; Morris, J. Ll.

doi:10.1090/S0025-5718-1975-0371085-X

Explicit-implicit schemes for the numerical solution of nonlinear hyperbolic systems
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by G. R. McGuire and J. Ll. Morris PDF

Math. Comp. 29 (1975), 407-424 Request permission

Abstract:

A class of methods, comprising combinations of explicit and implicit methods, for solving systems of conservation laws in one space dimension is developed. The explicit methods of McGuire and Morris [5] are combined with the implicit methods of McGuire and Morris [11] in a manner similar to that for creating Hopscotch methods (Gourlay [13]). The stability properties of these explicit-implicit methods is investigated and the results of some numerical experiments are presented. Extensions of these methods to systems of conservation laws in two space dimensions are also briefly discussed.

References

A. Jeffrey and T. Taniuti, Non-linear wave propagation. With applications to physics and magnetohydrodynamics, Academic Press, New York-London, 1964. MR 0167137
O. A. Oleĭnik, On discontinuous solutions of non-linear differential equations, Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 1098–1101 (Russian). MR 0083088

A Survey of Difference Methods for Non-Steady Fluid Dynamics

A. R. Gourlay and J. Ll. Morris, Finite difference methods for nonlinear hyperbolic systems, Math. Comp. 22 (1968), 28–39. MR 223114, DOI 10.1090/S0025-5718-1968-0223114-8
G. R. McGuire and J. Ll. Morris, A class of second-order accurate methods for the solution of systems of conservation laws, J. Comput. Phys. 11 (1973), 531–549. MR 331808, DOI 10.1016/0021-9991(73)90136-8
Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
Peter D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954), 159–193. MR 66040, DOI 10.1002/cpa.3160070112
Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
R. Courant, K. Friedrichs, and H. Lewy, On the partial difference equations of mathematical physics, IBM J. Res. Develop. 11 (1967), 215–234. MR 213764, DOI 10.1147/rd.112.0215
John Gary, On certain finite difference schemes for hyperbolic systems, Math. Comp. 18 (1964), 1–18. MR 158553, DOI 10.1090/S0025-5718-1964-0158553-3
G. R. McGuire and J. Ll. Morris, A class of implicit, second-order accurate, dissipative schemes for solving systems of conservation laws, J. Comput. Phys. 14 (1974), 126–147. MR 343641, DOI 10.1016/0021-9991(74)90009-6
S. Abarbanel and G. Zwas, An iterative finite-difference method for hyperbolic systems, Math. Comp. 23 (1969), 549–565. MR 247783, DOI 10.1090/S0025-5718-1969-0247783-2
A. R. Gourlay, Hopscotch: A fast second-order partial differential equation solver. , J. Inst. Math. appl. 6 (1970), 375–390. MR 0278537
A. R. Gourlay and G. R. McGuire, General hopscotch algorithm for the numerical solution of partial differential equations, J. Inst. Math. Appl. 7 (1971), 216–227. MR 287726
Paul Gordon, Nonsymmetric difference equations, J. Soc. Indust. Appl. Math. 13 (1965), 667–673. MR 185830

AIAA J.

A. R. Gourlay and J. Ll. Morris, Hopscotch difference methods for nonlinear hyperbolic systems, IBM J. Res. Develop. 16 (1972), 349–353. Mathematics of numerical computation. MR 347100, DOI 10.1147/rd.164.0349

One Dimensional Methods for the Numerical Solution of Nonlinear Hyperbolic Equations

John J. H. Miller, On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl. 8 (1971), 397–406. MR 300435

Hopscotch Methods for the Solution of Linear Second Order Parabolic Partial Differential Equations

Gilbert Strang, Accurate partial difference methods. II. Non-linear problems, Numer. Math. 6 (1964), 37–46. MR 166942, DOI 10.1007/BF01386051
Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI 10.1137/0705041

J. Computational Phys.

Patrick J. Roache, Computational fluid dynamics, Hermosa Publishers, Albuquerque, N.M., 1976. With an appendix (“On artificial viscosity”) reprinted from J. Computational Phys. 10 (1972), no. 2, 169–184; Revised printing. MR 0411358