High-order accurate and stable discretizations of partial differential equations with wave-like solutions

Abstract

Finally, numerical interface conditions are developed for an Auxiliary Differential Equation Generalized Dispersion Model (ADE-GDM) form of Maxwell’s equations. Compatibility conditions based on the primal interface conditions and the PDE are used to derive equations for ghost points to define the interface in the discretized problem. The method developed uses three time levels and a virtual step to maintain centered finite difference stencils. This ultimately provides better stability properties, allowing for large CFL-one like time steps for second and fourth order discretizations, so that the resulting scheme is efficient with respect to both time and memory complexity. A model problem analysis that focuses on a one dimensional dispersive wave equation with interface is used. The continuous interfaceconditions for the model problem are shown to be well-posed. A modal analysis is used to show that the set of semidiscrete interface conditions for the model problem are stable provided that each domain individually is stable. A matrix stability analysis and numerical results verify the accuracy and stability properties of the fully discrete interface conditions.

Wave equations are a class of partial differential equations (PDEs) that provide useful models for a variety of physical phenomena. Numerical methods for wave equations must carefully address stability due to energy conservation of the continuous equations, and high order approximations are useful for achieving greater efficiency. This dissertation considers the construction of high order accurate and stable numerical methods for a variety of wave equations. First, the Galerkin Difference Method, a finite element method based on a novel finite element space and corresponding set of basis functions that provide favorable properties, is adapted to use a discontinuous finite element space. The resulting discontinuous basis functions are used as the foundation of discontinuous Galerkin and interior penalty discretizations of first and second order hyperbolic PDEs. Upwind fluxes are used to introduce dissipation into the schemes, guaranteeing energy stability and providing favorable qualitative properties for lower regularity solutions. Since the second order wave equation is solved in its native second order form, an appropriate upwind flux must be derived from the solution to a related Riemann problem. Next, the original continuous basis functions are incorporated into interior penalty methods to provide systematic discretization of high order differential operators using nonconforming methods. In all cases, the variational forms are symmetrized in order to recover the superconvergence of the schemes observed in previous literature on Galerkin Difference methods. Analyses, including standard finite element analyses and grid dispersion analyses, and numerical results are presented which demonstrate the properties of schemes.