Implicit schemes and decompositions

Authors:
A. Jameson and E. Turkel

Journal:
Math. Comp. **37** (1981), 385-397

MSC:
Primary 65M10; Secondary 65F05

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628702-9

MathSciNet review:
628702

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Implicit methods for hyperbolic equations are analyzed by constructing *LU* factorizations. It is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the *LU* factors are diagonally dominant. Stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in *n* space dimensions. Only two factors are required even in three space dimensions. Acceleration to a steady state is analyzed. When the multidimensional backward Euler method is used with large time steps, it is shown that the scheme approximates a Newton-Raphson iteration procedure.

**[1]**R. W. Beam & R. F. Warming, "An implicit finite difference algorithm for hyperbolic systems in conservation form,"*J. Comput. Phys.*, v. 23, 1976, pp. 87-110.**[2]**W. R. Briley and H. McDonald,*Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique*, Proceedings of the Fourth International Conference on Numerical Methods in Fluid Dynamics (Univ. Colorado, Boulder, Colo., 1974) Springer, Berlin, 1975, pp. 105–110. Lecture Notes in Phys., Vol. 35. MR**0464908****[3]**J. A. Desideri, J. L. Steger & J. C. Tannehill,*On the Iterative Convergence Properties of an Implicit Approximate Factorization Finite Difference Algorithm*, Iowa State Univ. Engrg. Res. Inst. Rep. ERI-79049, 1978.**[4]**David Gottlieb, Max Gunzburger, and Eli Turkel,*On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods*, SIAM J. Numer. Anal.**19**(1982), no. 4, 671–682. MR**664877**, https://doi.org/10.1137/0719047**[5]**Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström,*Stability theory of difference approximations for mixed initial boundary value problems. II*, Math. Comp.**26**(1972), 649–686. MR**0341888**, https://doi.org/10.1090/S0025-5718-1972-0341888-3**[6]**Amiram Harten and Hillel Tal-Ezer,*On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I. Nonstiff strongly dynamic problems*, Math. Comp.**36**(1981), no. 154, 353–373. MR**606501**, https://doi.org/10.1090/S0025-5718-1981-0606501-1

A. Harten and H. Tal-Ezer,*On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. II. Five-point schemes*, J. Comput. Phys.**41**(1981), no. 2, 329–356. MR**626615**, https://doi.org/10.1016/0021-9991(81)90100-5**[7]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[8]**I. Lindemuth & J. Killeen, "Alternating direction implicit techniques for two dimensional magnetohydrodynamic calculations,"*J. Comput. Phys.*, v. 13, 1973, pp. 181-208.**[9]**D. H. Rudy & R. S. Hirsh,*Comments on the Role of Diagonal Dominance in Implicit Difference Methods*, NASA Tech. Memo, NASA TM-X-73905, 1976.

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10,
65F05

Retrieve articles in all journals with MSC: 65M10, 65F05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628702-9

Article copyright:
© Copyright 1981
American Mathematical Society