On the stability of block difference schemes for parabolic problems. (English. Russian original) Zbl 0662.65049
Sov. Math., Dokl. 36, No. 3, 489-492 (1988); translation from Dokl. Akad. Nauk SSSR 297, 531-534 (1987).
The authors investigate a discretization (in t only) for the abstract Cauchy problem: \(u'(t)+Au(t)=f(t),\) \(0\leq t\leq 1\), \(u(0)=\phi\) in a Banach space E, where A is an unbounded linear operator with a compact domain D(A) in E (in the applications A is an elliptic differential operator in the space variables). The theory of the method of block approximation and the theory of stability is developed. Coercivity inequalities for difference schemes are established. Four theorems are proved.
Reviewer: I.Grosu
MSC:
| 65J10 | Numerical solutions to equations with linear operators |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 34G10 | Linear differential equations in abstract spaces |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
