Bertil Gustafsson
Department of Scientific Computing
Uppsala University
Sweden

“Deferred correction methods for initial-boundary value problems”

Abstract

The deferred correction method, originally presented by Fox and further developed by Pereyra, is an efficient way of obtaining high order accuracy for boundary value problems. A method of order p is constructed by solving a sequence of p/2 systems based on a symmetric second order difference approximation with different right hand sides. The advantage is particularly pronounced, when considering domain decomposition methods, since the compact form of the second order approximation introduces minimal coupling between the sub-domains.

The method can also be used for initial-boundary value problems. There are essentially three different approaches for such problems. One can use the deferred correction principle in space only, in time only, or simultaneously in space and time. In earlier work we have developed such methods using the first two approaches, and currently, we are working on the third type of approach.

In this talk we will first give an overview of the method, and present some theoretical and experimental results for the initial-boundary value problem. By using the trapezoidal rule as the basic scheme for discretization in time, we obtain order of accuracy p by applying the scheme p/2 times for each time-step (with different right hand sides). By slightly weakening the stability concept, one can show that the scheme is unconditionally stable.

Then we will present recent work on the deferred correction principle in space and time. In particular, we will discuss the wave equation in second order formulation as well as in first order system formulation, where in the latter case a staggered grid is used. If an explicit second order standard scheme is used as the basic approximation, there is a stability condition on the time-step. By working with the right stability concept, the time-step is not further restricted for our higher order deferred correction scheme, as is the case for other types of explicit higher order approximations.