Difference Schemes for Nonstationary Vector Problems

Abstract

So far, twoand three-level difference schemes have been constructed and investigated for basic nonstationary problems of mathematical physics [1, 2]. We mainly speak of initial-boundary value problems for a second-order parabolic equation (the heat equation). In this case, the elliptic operator with respect to the space variables may be self-adjoint or nonself-adjoint, as in the convection–diffusion equation [3]. Similar results can be obtained for second-order hyperbolic equations (the oscillation equation). In the nonself-adjoint case, unconditionally stable schemes are most often constructed under the condition that the nonself-adjoint part of the operator is subordinate to the self-adjoint part [2]. These results can be generalized in the theory of operator-difference schemes, where operator statements in various spaces for the space derivatives are used in the initial-boundary value problems of mathematical physics [1, 2]. This permits one to consider various classes of nonstationary problems of mathematical physics from a common viewpoint by neglecting, in particular, the specific nature of differential equations and boundary conditions. Results on stability with respect to the initial data and the right-hand side and stability under perturbations of the problem operator (the coefficients of the original equation) were obtained in [4] for twoand three-level operatordifference schemes. In the approximate solution of initial-boundary value problems for multidimensional partial differential equations, special attention is paid to the construction of additive schemes, that is, decomposition schemes [1, 5, 6]. The passage to a chain of simpler problems permits one to construct efficient difference schemes (decomposition with respect to the space variables). In a number of cases, it is useful to extract subproblems of different nature (decomposition with respect to physical processes). Regionally additive schemes (domain decomposition schemes) designed for the construction of numerical algorithms for parallel computers [6] have been actively discussed in recent time. In the case of a multicomponent decomposition (into three or more operators), unconditionally stable additive schemes are usually constructed with the use of the notion of total approximation on the basis of the passage to a chain of separate initial value problems for each operator term. In a number of cases, additive schemes of multicomponent decomposition are constructed without using the total approximation (so-called regularized additive schemes [6]). We also single out a new class of additive schemes for vector problems. Such schemes can be used in the construction of effective numerical algorithms for the approximate solution of systems of equations of mathematical physics, that is, vector equations. Typical is the situation in which separate components of the vector of unknowns are coupled with each other, and it is difficult to pose a good problem for the components on a new time level. Note some examples of the construction of additive schemes for vector problems. In the application of additive difference schemes to systems of parabolic and hyperbolic equations in [1], the passage to simpler problems is performed on the basis of the regularization principle. The regularization principle was suggested in [7], where its capabilities were illustrated by a number of meaningful examples. The most impressive results [1, 2] were obtained in the construction of unconditionally stable difference schemes for nonstationary problems of mathematical physics, factorized schemes for multidimensional problems, and iterative methods for grid problems [8]. One improves the properties of a difference scheme by perturbing the operators of the