Difference Schemes of High Accuracy Order on Uniform Grids for a Singularly Perturbed Parabolic Reaction-Diffusion Equation

Abstract

For a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon$ ($\varepsilon \in (0,1]$) multiplying the highest-order derivative, we consider a technique to construct $\varepsilon$-uniformly convergent in the maximum norm difference schemes of higher accuracy order on uniform grids. In constructing such schemes, we use the solution decomposition method, in which grid approximations of the regular and singular components in the solution are considered on uniform grids. Increasing of the convergence rate of the scheme constructed with improved accuracy of order $\mathcal{O}\left (N^{-4}\,\ln ^{4}\,N + N_{0}^{-2}\right )$, where N and N ₀ are the number of nodes in the meshes in x and t, respectively, is achieved using a Richardson extrapolation technique applied to the regular and singular components. In the proposed Richardson technique, when constructing embedded grids we use most dense grids as main grids. This approach allows us to construct schemes that converge $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal{O}\left (N^{-6}\,\ln ^{6}\,N + N_{0}^{-3}\right )$ and higher.