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A convergence analysis of hopscotch methods for fourth order parabolic equations. (English) Zbl 0608.65078

The authors consider the recursion \((I+C)U_{n+1}=BU_ n+(I+C)U_{n- 1}\) arising when a hopscotch method is applied to a fourth-order parabolic equation. The matrices B, C are real skew-Hermitian. A condition for the method to be stable is derived. It is not clear whether this condition is best possible or whether it holds when the coefficients of the original problem vary in space and time. In fact the well-known energy method yields all this information.
Reviewer: J.D.P.Donnelly

MSC:

65N40 Method of lines for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations

Software:

Hopscotch
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References:

[1] Evans, D.J., Danaee, A.: A new group hopscotch method for the numerical solution of partial differential equations. SIAM J. Numer. Anal.19, 588-598 (1982) · Zbl 0486.65061 · doi:10.1137/0719039
[2] Gane, C.R., Gourlay, A.R.: Block hopscotch procedures for second order parabolic differential equations. JIMA19, 205-216 (1977) · Zbl 0356.65087
[3] Gourlay, A.R.: Hopscotch: A fast second-order partial differential equation solver. JIMA6, 375-390 (1970) · Zbl 0218.65029
[4] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0148.12601
[5] ter Maten, E.J.W.: Stability analysis of finite difference methods for fourth order parabolic partial differential equations. Thesis. University of Utrecht, Utrecht 1984
[6] ter Maten, E.J.W., Sleijpen, G.L.G.: Hopscotch methods for fourth order parabolic equations I: stability results for fixed stepsizes. Preprint 275, Mathematical Institute, University of Utrecht, Utrecht 1983
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