Temporal and Pseudo-Temporal Numerical Integration Methods

Abstract

Numerical methods for integrating partial differential equations are used in nearly every scientific field. In this dissertation we study two types of numerical integration methods, transient methods and pseudo-transient methods. Transient methods for partial differential equations look for time-accurate solutions that explain the evolution of the equation (although a steady state solution may evolve). Pseudo-transient methods look for steady-state solutions of partial differential equations while paying attention to the transient behavior to aid in stability. In contrast, root-finding methods, e.g. line-search methods, look only for a steady-state solution often not paying attention at all to the transient behavior of the problem.

Pseudo-transient continuation is a method for solving steady state solutions of partial differential equations, and is used when traditional line-search methods fail to converge or converge to non-physical solutions. The method is a hybrid between implicit Euler and Newton's method where variable step-sizes are used to transfer from one method to the other. We demonstrate the performance of pseudo-transient continuation both numerically and theoretically on a variety of problems. We extend the global convergence theory, which currently covers a class of ordinary differential equations, to include a class of semi-explicit index-1 differential-algebraic equations.

We also studied CVode, a transient code for solving nonlinear partial differential equations. In this work, we explain how CVode was extended to allow for a two-grid nonlinear solver. The two-grid solver coarsens a given mesh and solves the nonlinear problem on the coarse mesh, which is then moved back to the fine mesh for refining. This approach can be less expensive than computing the full nonlinear solution on the fine mesh. We explore some of the theoretical and computational issues involved in implementing this method for a radiative transfer problem as might be seen in stellar fusion.