The acceleration of the Peaceman-Rachford method by Chebyshev polynomials

Abstract

Considerable interest centres at the present time on the application of alternating direction implicit (ADI) procedures to the numerical solution of elliptic systems of equations (see for example the excellent book of Wachspress (1966)). Under model problem conditions such methods have very rapid convergence rates. It is the purpose of this note to show how such convergence rates may be improved on by building into the ADI method an acceleration procedure based on the use of Chebyshev polynomials. In Section 2, the Chebyshev semi-iterative procedure is summarized; a fuller account may be found in Varga (1962). In Section 3, the ADI process is denned. Section 4 contains the new process applied to an ADI procedure with a constant acceleration parameter, whilst in Section 5 this new process is generalized to the important case when the ADI procedure has a cycle of acceleration parameters. An example is given in Section 6 where this process is applied to solving Laplace's equation by two well-known methods. In Section 7 a more efficient version of the procedure outlined in Section 4 is derived. This procedure is of general value.