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 (l1.1) Ax = b, where Di B1 x b1 C2 D2 B2 2b 0 (1.2) A= C3 D3 B3 , , 0 BM-1 CM DM _ XMJ b M and where the Di are N x N tridiagonal matrices, the Bi and Ci are appropriate diagonal or tridiagonal matrices, and x and b are vectors partitioned accordingly. (For simplicity we assume C1 and Bi are N x N.) Assume the matrix is nonsymmetric and the corresponding difference equation is nonseparable. We propose a marching technique to solve the above. The marching technique can be described briefly as follows. Consider the nth equation block; i.e., (1.3) Cnxn-1 + Dnxn + Bnxn + = bnReceived April 17, 1978. AMS (MOS) subject classifications (1970). Primary 65N20, 65F05, 15A06. *This work was supported by the Department of the Navy, Naval Sea Systems Command, under Contract No. N00024-78-C-5384. 1979 American Mathematical Society 0025-571 8/79/0000-0101 /$03.50 881 This content downloaded from 157.55. 39.243 on Wed, 05 Oct 2016 04:16:15 UTC All use subject to http://about.jstor. org/terms 882 LOUIS W. EHRLICH Assuming we can solve for xn + 1, we have (1.4) = B C1x,b -D~x) (1.4) x~~n + 1 = n (n CnXn-1 -n Xn)' Thus, if we guess xl, then we can march along solving for the other xi The final equation (1.5) CMXM-l + DMxM = bM will in general not be satisfied. We, thus, must modify our guess for xi so that the last equation will be satisfied on a second march through the region. This modification is done by inverting an "influence matrix", in the terminology of Roache [11], [12]. Bank and Rose [2], [3] consider separable equations and look at the marching procedure as an LU decomposition of the given matrix. Further, the separability allows the influence matrix to be inverted via polynomial decomposition. Roache [11], [12], on the other hand, actually produces the influence matrix and uses a direct inversion scheme. Since we do not have separability, we use the latter. The influence matrix is actually produced by marching over the region with xi equal to all zeros except for a single unknown which is equal to 1. This is repeated for each of the N unknowns. The errors in XM, determined from (1.5), for each unknown then become the appropriate columns of the N x N influence matrix. Because marching techniques are inherently unstable [2], [11], we propose a multiple shooting method as used by Bank and Rose [2], [3], i.e., the marching is limited to blocks of q lines. Thus, we first permute the matrix so that we have [G1 1 G2 W2 V2 G3 I W3 V3 o Vk V-1 (1.6) AX= Gk wk