Conservative compact and monotone fourth order difference schemes for quasilinear equations

Abstract

In this work, for the first time, compact and monotone difference schemes of the 4th order of accuracy are constructed and studied, preserving the property of conservation (divergence), for a quasilinear stationary reaction-diffusion equation. To linearize the nonlinear difference scheme, an iterative method of the Newton-Seidel type is used, which also preserves the idea of conservation and monotonicity of the iteration. The main idea of implementing the proposed difference scheme on a three-point stencil of the sweep method is based on the possibility of parallelizing the computational process. First, the solution is at the even nodes, and then at the odd ones. In this case, all equations remain three-point with respect to the unknown function. The arising problems of finding additional boundary conditions at the boundary nodes are solved using the Newton interpolation polynomial of the 4th order of accuracy. The presented results of the computational experiment illustrate the effectiveness of the proposed algorithm. The possibility of generalizing this method to more difficult problems is also indicated.