| 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.;The second topic is stability for split methods. After a demonstration that in general the product of two stable operators need not be stable, some important classes of hyperbolic splittings are identified for which the product of stable approximate solution operators is in fact stable.;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. A procedure is described which, for many problems, can be used to transform the given boundary conditions for the original equation into arbitrarily accurate boundary conditions for the intermediate solutions. Stability of the initial-boundary value problem is also discussed.;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 of the theory. Two-dimensional hyperbolic problems, convection-diffusion equations, and the Peaceman-Rachford ADI method for the heat equation are considered.;Three main aspects of the method are considered. The first is the accuracy and efficiency of the time-split method relative to unsplit methods. We derive a general expression for the splitting error and use it to compute the overal truncation error for the time-split method. This is then used to analyze its efficiency, measured by the amount of work required to obtain a given accuracy. |
