Nonlinear difference schemes for the integration of multidimensional equations

Abstract

We describe the methods of constructing the difference schemes of an arbitrary-order approximation which ensure the validity of the maximum principle. The methods are based on the nonlinear correction of the schemes, while the requirement for the identity of the initial schemes with the corrected ones on smooth functions without extrema is met. We construct the algorithms for correcting the elliptic, parabolic, and hyperbolic equations.One of the ways to improve the difference schemes is to provide the properties at discrete level which are characteristic of the original equations. The examples are conservativeness and complete conservativeness (balance of internal energy) in gas dynamics problems, the monotonicity of the solution in one-dimensional problems for a transport equation, the validity of the maximum principle in some three-dimensional problems, the invariance of the differential approximations of the schemes under the transformations under which the original equations are invariant.The importance of the above properties depends on the classes of problems solved. In particular, the monotonicity of solutions and the maximum principle are mainly needed when integrating discontinuous solutions. However, they also allow us to improve the accuracy of calculations and smooth solutions when using difference grids with a small number of nodes. We should emphasize that the maximum principle may hold for schemes with constant coefficients if the counterflow formulae of a first-order approximation are used for the first derivatives, the symmetric three-point formulae for the second derivatives, which allows us to obtain schemes with positive coefficients. At the same time any homogeneous approximations of higher derivatives (if the approximations of the first or second derivatives do not prevail) do not generally ensure the validity of the maximum principle, though for the corresponding differential equations the maximum principle holds or the solutions remain monotonic.It is possible to overcome the above limitations, using the class of nonlinear schemes. Thus, the monotonic schemes with second-or higher-order approximation, which have the properties of adaptation to the 'smoothness' of the solution, are constructed for hyperbolic equations [1][2][3][4][5]11,12]. In particular, the general algorithms for the monotonizing correction of arbitrary schemes in canonical form are proposed in [4,5]. The generalization of these algorithms is given in [6][7][8]. They allow us to construct adaptive schemes for solving one-dimensional parabolic equations (the Cauchy problem) or a boundary value problem for ordinary equations. Analogous methods are employed here to construct nonlinear schemes for solving equations with two space variables, and all the results obtained hold for the case of an arbitrary number of variables. We consider two-dimensional problems only for brevity.