Article
MSC:
34G10,
35G10,
35K25,
65J10,
65L05,
65M10,
65N12 | MR 0695181 | Zbl 0538.65032 | DOI: 10.21136/AM.1983.104008
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Keywords:
unconditional stability; complex Banach space; finite difference method; $k$-step formula
unconditional stability; complex Banach space; finite difference method; $k$-step formula
Summary:
The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the $A_n$-acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary differential equations as well.
References:
[1] I. Babuška M.Práger, E. Vitásek: Numerical Processes in Differential Equations. SNTL, Prague, 1966. MR 0223101 | Zbl 0156.16003
[2] B. L. Ehle: A-stable Methods and Pade Approximations to the Exponential. SIAM J. Math. Anal., Vol. 4 (1973), No. 4, pp. 671-680. DOI 10.1137/0504057 | MR 0331787 | Zbl 0236.65016
[3] A. Iserles: On the A-acceptability of Pade Approximations. SIAM J. Math. Anal., Vol. 10 (1979), No. 5, pp. 1002-1007. DOI 10.1137/0510091 | MR 0541096 | Zbl 0441.41010
[4] E. Hille, R. S. Phillips: Functional Analysis and Semi-groups. Amer. Math. Soc., Vol. 31., rev. ed., Waverly Press, Baltimore, 1957. MR 0089373
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