A domain decomposition method for solving a two-dimensional Viscous Burger's Equation

Abstract

We present an efficient finite difference numerical solution for a convection equation that includes diffusion terms with small coefficients. The equation is first advanced on a coarse mesh. Regions of significant diffusion activity are identified using a threshold criterion. The coarse mesh is dynamically decomposed and the mesh is refined in these regions. The decomposition is heterogeneous in the sense that the problem formulations as well as the solution methods may vary from mesh to mesh. In particular, a mixed Euler-Lagrange method is developed that explicitly advances the coarse mesh relative to an Eulerian reference frame, and implicitly advances the refined meshes relative to moving Lagrangian reference frames. On the refined meshes the method is second-order accurate in space. Asynchronous Schwarz iterations are used between overlapping refined meshes, when needed, to communicate the data dependence of the implicit refined solution. The computation is distributed across the four processors of the shared memory CRAY-2.