Local uniform grid refinement and systems of coupled partial differential equations

Abstract

In this paper we consider an adaptive grid method based on local uniform grid refinement applied to systems of coupled time-dependent PDEs. Local uniform grid refinement means that the PDEs are solved on a series of nested, uniform, increasingly finer subgrids which cover only a part of the domain. These subgrids are created up to a level of refinement where sufficient spatial accuracy is obtained and their location and shape is adjusted after each time step in order to follow the moving steep fronts. When a system of coupled PDEs is solved, the behavior of the local and global error associated with each separate PDE can be very different from one PDE to another. A refinement strategy based on a global error analysis has been developed which takes these differences into account. This refinement strategy aims at the domination of the global space error by the space discretization error at the finest subgrid.