Implicit schemes and decompositions
Authors:
A. Jameson and E. Turkel
Journal:
Math. Comp. 37 (1981), 385397
MSC:
Primary 65M10; Secondary 65F05
MathSciNet review:
628702
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Abstract 
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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 NewtonRaphson iteration procedure.
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 R. W. Beam & R. F. Warming, "An implicit finite difference algorithm for hyperbolic systems in conservation form," J. Comput. Phys., v. 23, 1976, pp. 87110.
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 W. R. Briley & H. McDonald, Solution of the Three Dimensional Compressible NavierStokes Equations by an Implicit Technique, Proc. Fourth Internat. Conf. Numerical Methods in Fluid Dynamics, Lecture Notes in Phys., vol. 35, SpringerVerlag, New York, 1974, pp. 105110. MR 0464908 (57:4827)
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 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. ERI79049, 1978.
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 D. Gottlieb, M. Gunzburger & E. Turkel, "On numerical boundary treatment of hyperbolic systems for finite difference and finite element methods," SIAM J. Numer. Anal. (To appear.) MR 664877 (83f:65154)
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 B. Gustafsson, H. O. Kreiss & A. Sundström, "Stability theory of difference approximations for mixed initial boundary value problems. II," Math. Comp., v. 26, 1972, pp. 649686. MR 0341888 (49:6634)
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 D. H. Rudy & R. S. Hirsh, Comments on the Role of Diagonal Dominance in Implicit Difference Methods, NASA Tech. Memo, NASA TMX73905, 1976.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106287029
PII:
S 00255718(1981)06287029
Article copyright:
© Copyright 1981
American Mathematical Society
