Interpolation finite difference schemes on grids locally refined in time

Abstract

Numerical methods in which the mesh is locally refined are widely used for problems with singularities in the solution. In this case, approaches with refining the grid in both space and time are being developed. In this paper, we consider a class of finite difference schemes with local refinement of the grid in time to solve the problems numerically; here we compute the numerical solution on a finer time grid in a part of the domain. We consider a model Dirichlet problem for a second-order parabolic equation on a rectangle. We analyze the accuracy of completely implicit schemes with the simplest interpolated interface conditions on the boundary of the adaptation domain. On the basis of the maximum principle, the unconditional convergence of these schemes in the uniform norm is shown, and the rate of convergence is analyzed.