Two-dimensional splitting schemes for hyperbolic equations

Abstract

The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional (p≥3 ) hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G при p=3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region G can be a connected union of cylindrical regions with generators parallel to the axis OX₃. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is constructed, which is a chain «two-dimensional problem – one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). The stability and convergence of the constructed schemes are proved: with the factorized rate O(ǁhǁ²+τ²), and with the additive rate O(ǁhǁ²+τ) , where ǁhǁ is the norm of the step of the spatial grid,, τ is the time step, under the appropriate restrictions on the smoothness of the functions included in the statement of the initial-boundary value problem. For the numerical implementation of the constructed schemes – the numerical solution of two-dimensional elliptic problems – one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.