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 shar p spatial transitions. Using nested, finer-and-finer uniform subgrids, the LUGR technique refines the space grid locally around these transi tions, so as to avoid discretization on a very fine grid covering the entire physical domain. This paper examines the LUGR technique for tim e-dependent problems when combined with static regridding. Static regr idding 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. Follo wing the method of lines approach, we develop a mathematical framework for the general Runge-Kutta LUGR method applied to multispace-dimensi onal problems. We hereby focus on parabolic problems, but a considerab le part of the examination applies to hyperbolic problems as well. Muc h attention is paid to the local error analysis. The central issue her e is a ''refinement condition'' which is to underly the refinement str ategy. By obeying this condition, spatial interpolation errors are con trolled in a manner that the spatial accuracy obtained is comparable t o the spatial accuracy on the finest grid if this grid would be used w ithout any adaptation. A diagonally implicit Runge-Kutta method is dis cussed for illustration purposes, both theoretically and numerically.