Two-Level Finite Difference Scheme of Improved AccuracyOrder for Time-Dependent Problems of Mathematical Physics

摘要

In the theory of finite difference schemes, the most complete results concerning the accu-racy of approximate solutions are obtained for two- and three-level finite difference schemes that con-verge with the first and second order with respect to time. When the Cauchy problem is numericallysolved for a system of ordinary differential equations, higher order methods are often used. Using amodel problem for a parabolic equation as an example, general requirements for the selection of thefinite difference approximation with respect to time are discussed. In addition to the unconditionalstability requirements, extra performance criteria for finite difference schemes are presented and theconcept of SM stability is introduced. Issues concerning the computational implementation ofschemes having higher approximation orders are discussed. From the general point of view, variousclasses of finite difference schemes for time-dependent problems of mathematical physics are ana-lyzed.