A Marching Technique for Nonseparable Equations

A multiple-shooting marching technique is described which is applicable to arbitrary block tridiagonal matrices derived from nonseparable difference equations which are solved many times. Comparison with other methods on a particular problem shows the method to be competitive with respect to time and storage. Introduction. Our interest here is in solving the difference equation which results from approximating a nonseparable partial differential equation. There are a variety of techniques available; and if our problem is only to be solved a few times, most any method will do. However, if the system is to be repeatedly solved many times (such as the stream function equation in a time dependent Navier-Stokes problem), our interest then centers on fast, efficient methods consistent with our storage availability. We present here a multiple shooting marching method which appears to be competitive in time, consistent with limited storage, with other available methods. The method is applied to a particular problem and compared with other techniques. 1. The Marching Technique. Consider the linear system (1.1) Ax = b, where (1.2) A = D, B, D, B, D, B, BM-i -JIÍ DM X =