Runge-kutta Methods and Local Uniform Grid Refinement

Local uniform grid refinement (LUGR) is an adaptive grid technique for computing solutions of partial differential equations possessing sharp spatial transitions. Using nested, finer-and-finer uniform subgrids, the LUGR technique refines the space grid locally around these transitions, so as to avoid discretization on a very fine grid covering the entire physical domain. This paper examines the LUGR technique for time-dependent problems when combined with static regridding. Static regridding means that in the course of the time evolution, the space grid is adapted at discrete times. The present paper considers the general class of Runge-Kutta methods for the numerical time integration. Following the method of lines approach, we develop a mathematical framework for the general Runge-Kutta LUGR method applied to multispace-dimensional problems. We hereby focus on parabolic problems, but a considerable part of the examination applies to hyperbolic problems as well. Much attention is paid to the local error analysis. The central issue here is a "refinement condition" which is to underly the refinement strategy. By obeying this condition, spatial interpolation errors are controlled in a manner that the spatial accuracy obtained is comparable to the spatial accuracy on the finest grid if this grid would be used without any adaptation. A diagonally implicit Runge-Kutta method is discussed for illustration purposes, both theoretically and numerically.