Reduction of a three-layer semi-discrete scheme for an abstract parabolic equation to two-layer schemes. Explicit estimates for the approximate solution error. (English. Russian original) Zbl 1320.65133
J. Math. Sci., New York 206, No. 4, 424-444 (2015); translation from Sovrem. Mat. Prilozh. 89 (2013).
Summary: We consider a purely implicit three-layer semi-discrete approximation scheme of second order for an approximate solution of the Cauchy problem for an abstract parabolic equation. Using the perturbation algorithm, we reduced this scheme to two two-layer schemes. The solutions of these schemes are used for the construction of an approximate solution of the initial problem. Explicit estimates for the approximate solution error are proved using the properties of a semi-group. To illustrate the generality of the perturbation algorithm when it is applied to difference schemes, a four-layer scheme reduced to two-layer schemes is also considered.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35K90 | Abstract parabolic equations |
| 65J08 | Numerical solutions to abstract evolution equations |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
semidiscretization; error estimate; Cauchy problem; abstract parabolic equation; perturbation algorithm; two-layer schemes; difference schemeReferences:
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