Time-Split Methods for Partial Differential Equations.
by Randall J. LeVeque. PhD Dissertation, Stanford University, 1982.

Abstract. This thesis concerns the use of time-split methods for the numerical solution of time-dependent partial differential equations. Frequently the differential operator splits additively into two or more pieces such that the corresponding subproblems are each easier to solve than the original equation, or are best handled by different techniques. In the time-split method the solution to the original equation is advanced by alternately solving the subproblems. In this thesis a unified approach to splitting methods is developed which simplifies their analysis. Particular emphasis is given to splittings of hyperbolic problems into subproblems with disparate wave speeds. Three main aspects of the method are considered. The first is the accuracy and efficiency of the time-split method relative to unsplit methods. The second topic is stability for split methods. The final topic is the proper specification of boundary data for the intermediate solutions, e.g., the solution obtained after solving only one of the subproblems. The main emphasis is on hyperbolic problems, and the one-dimensional shallow water equations are used as a specific example throughout. The final chapter is devoted to some other applications or the theory. Two-dimensional hyperbolic problems, convection-diffusion equations, and the Peaceman-Rachford ADI method for the heat equation are considered.

Keywords. *Partial differential equations, *Time dependence, *Splitting, *Numerical methods and procedures, *Solutions(General), Accuracy, Efficiency, Stability, Boundary value problems, Hyperbolas, Cauchy problem, Operators(Mathematics), Two dimensional, One dimensional, Shallow water, Waves, Velocity, Theses

bibtex entry:

```@phdthesis{leveque:phd,
Author = {R. J. {LeVeque}},
School = {Stanford University},
Title = {Time-split methods for partial differential equations},
Year = {1982}}
```