Analysis of local uniform grid refinement

Abstract

Numerical methods for time-dependent PDEs usually integrate on a fixed grid, a priori chosen for the whole time interval. Similar to a fixed stepsize, a fixed grid may be inefficient when solutions possess large local gradients. While most schemes can easily adapt the stepsize, as in genuine ODE and method-of-lines schemes, the question of how to automatically adapt the grid to rapid spatial transitions is much more involved. The subject of this paper is local uniform grid refinement (LUGR) for finite different methods. The idea of LUGR is to cover the spatial domain with nested, finer-and-finer, locally uniform subgrids. LUGR is applicable both to stationary and time-dependent problems. For time-dependent problems the local subgrids are adapted at discrete values of time to follow moving transitions. The aim of this paper is to discuss, for the class of finite difference methods under consideration, a general error analysis that shows the interplay between local truncation and interpolation errors. This analysis points the way to a theoretically optimal strategy for the local refinement, optimal in the sense that this strategy controls accumulation of interpolation errors and simultaneously strives for the spatial accuracy that would be obtained on the finest grid when used without adaptation. Attention is paid to both the stationary and time-dependent case, while for time-dependent problems the emphasis lies on combining LUGR with Runge-Kutta time stepping.