ADAPTIVE MESH REFINEMENT FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

In many time dependent simulations, the solution on most of the domain will be fairly smooth, with discontinuities or highly oscillatory phenomena occurring over only a small fraction of the domain. In problems such as these, a mesh refinement approach can be the most efficient, and often the only practical, solution method. Refined grids with smaller and smaller mesh spacing are placed only where they are needed. Since we are solving a time dependent problem, the regions needing refinement will change, and therefore our grids must adapt with time as well.; This thesis presents a method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations (pde) using explicit finite difference techniques. Based upon Richardson-type estimates of the local truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. In addition, this approach is recursive in that fine grids can themselves contain even finer subgrids. Those grids with finer mesh width in space will also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space.; In chapter 2 we present the full mesh refinement algorithm. This includes a discussion of the error estimation procedure, as well as the integration algorithm itself. We calculate how often the error estimation and subsequent grid generation should be performed to minimize the total cost of the algorithm.; In chapter 3 we discuss the interface equations used at the boundary of the fine and coarse grids. Boundary schemes are presented for those component grids with boundaries internal to the problem domain, since in these cases boundary conditions are not supplied with the pde. We present a new stability proof for the case of interpolation interface conditions with the Lax-Wendroff difference scheme for mesh refinement in time and space.; We also include in this chapter a discussion of conservation properties of the algorithm, including both the interface conditions and the regridding procedure. We develop a procedure to derive conservative interface conditions so that if the approximate solutions converge, they will converge to a weak solution of the pde. This is important in computing the correct shock location in cases where the solution is discontinuous. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI...