Generalization of the Lax–Ryabenky–Philippov theorem to nonlinear problems
https://doi.org/10.29235/1561-8323-2021-65-4-391-396
Abstract
In this paper, Lax’s equivalence theorem, which states that stability is a necessary and sufficient condition for its convergence in the presence of an approximation of a difference scheme, is generalized to abstract nonlinear difference problems with operators acting in finite dimensional Banach spaces. In contrast to linear finite-difference methods, such a criterion in the nonlinear case can be established only for unconditionally stable computational methods, when the corresponding a priori estimates take place for sufficiently small |h| ≤ h0. In this case, the value of h0 depends both on the consistency of discrete and continuous norms in Banach spaces, and on the magnitude of the perturbation of the input data of the problem. The proven convergence criterion is used to study the stability of difference schemes approximating quasilinear parabolic equations with nonlinearities of unbounded growth with respect to the initial data.
About the Author
P. P. Matus
Institute of Mathematics of the National Academy of Sciences of Belarus; Institute of Mathematics and Computer Science the John Paul II Catholic University of Lublin
Belarus
Belarus
Matus Piotr P. – Corresponding Member, D. Sc. (Physics and Mathematics), Professor, Chief researcher
11, Surganov Str., 220072, Minsk
Lublin
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