Conservative Compact and Monotone Fourth-Order DifferenceSchemes for One- and Two-Dimensional Quasilinear Equations
Abstract
Compact and monotone difference schemes of the fourth order of accuracy preserving the conservativeness (divergence) property for the one- and two-dimensional quasilinear stationary reaction-diffusion equations are constructed and investigated. A priori estimates of the difference solution in the nonlinear case for the one-dimensional quasilinear equation are obtained based on the established two-sided estimates of the grid solution. For the linearization of the nonlinear difference scheme, an iterative method of the Newton–Seidel type preserving conservativeness and monotonicity is used. The main idea of the proposed difference schemes is based on the possibility of parallelizing the computational process. The emerging problems of finding additional boundary conditions at boundary nodes in both one- and two-dimensional cases are solved using the Newton interpolation polynomial of the fourth order of accuracy. The presented results of the computational experiments illustrate the increased order of the proposed algorithms. The possibility of generalizing this method to nonstationary quasilinear equations is also indicated.