ANALYSIS OF LOCAL UNIFORM GRID REFINEMENT

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 larg e local gradients. While most schemes can easily adapt the stepsize, a s in genuine ODE and method-of-lines schemes, the question of how to a utomatically 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 s patial 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 va lues of time to follow moving transitions. The aim of this paper is to discuss, for the class of finite difference methods under considerati on, a general error analysis that shows the interplay between local tr uncation and interpolation errors. This analysis points the way to a t heoretically optimal strategy for the local refinement, optimal in the sense that this strategy controls accumulation of interpolation error s and simultaneously strives for the spatial accuracy that would be ob tained on the finest grid when used without adaptation. Attention is p aid to both the stationary and time-dependent case, while for time-dep endent problems the emphasis lies on combining LUGR with Runge-Kutta t ime stepping.