AN ADAPTIVE MESH ALGORITHM FOR SOLVING SYSTEMS OF TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS (HYPERBOLIC, MOVING, REFINEMENT)

We discuss an adaptive mesh algorithm that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time dependent partial differential equations in two space dimensions. Our algorithm combines the adaptive techniques of mesh moving, static rezoning, and local mesh refinement. The nodes of a coarse mesh of quadrilateral cells are moved by a simple algebraic node movement function, determined from the geometry and propagation of regions having statistically significant discretization error or mesh movement indicators. The local mesh refinement method recursively divides cells of the moving coarse mesh within clustered regions that contain nodes with large error until a user prescribed error tolerance is satisfied. These finer grids are properly nested within the moving coarse mesh to provide for simpler data structures and interface conditions between the fine and coarse meshes.; Our procedure is designed to be flexible, so that it can be used with many existing finite difference and finite element schemes and with different error estimation procedures. To test our adaptive mesh algorithm, we implemented it in a system code with an initial mesh generator, a MacCormack finite difference scheme for hyperbolic vector systems of conservation laws, and a Richardson extrapolation based error estimation. Results are presented for several computational examples.; The moving mesh technique reduces dispersive errors near shocks and wave fronts. Therefore, it reduces the grid requirements necessary to compute accurate solutions and thus increases computational efficiency. The local mesh refinement provides smaller mesh spacings and time steps in regions where the problem is difficult to solve, thus providing increased accuracy and enabling error tolerances to be achieved.