Efficient finite difference schemes for wave equations: part 1: incompressible linear elasticity part 2: hierarchical high-order accurate schemes

Abstract

This thesis is comprised of two major contributions. The first is the formulation of an efficient finite difference scheme for time-dependent incompressible linear elasticity on complex geometry. The governing equations are solved in displacement-pressure form to second-order accuracy in space and time. A fractional-step approach is taken so that the time update for the displacement is performed separately from the solution to the Poisson equation for the pressure. Overset grids are employed to effectively describe interfaces and physical boundaries for complex geometries. A particular form of upwind dissipation is included in the scheme to ensure stability on overset grids and traction boundaries. Divergence damping is added into the scheme to maintain small dilatations. A Gustafsson, Kreiss, and Sundstrom (GKS) normal mode analysis is performed on a model problem to verify the stability of this scheme with displacement and traction boundary conditions. To showcase the accuracy and stability of the scheme, several numerical experiments using known solutions for varying geometries are shown. The second main result is the development of a novel framework for the construction of high-order accurate finite difference schemes for the wave equation, Maxwell's equations, and incompressible linear elasticity. The framework consists of a hierarchy in which high-order accurate approximations are formed from lower-order ones.Each level within this hierarchy is constructed using a second-order accurate finite difference scheme. To obtain higher-order accuracy, the second-order scheme is augmented with corrections which are found according to a modified-equation approach. In addition, higher-order accurate discrete boundary and interface conditions are constructed as part of this hierarchy. The overall schemes use only three time levels and are CFL-one stable. Application of von Neumann analysis shows the stability of these high-order accurate hierarchical schemes and numerical experiments verify the predicted stability and accuracy.