Optimized block-implicit relaxation

• Published in: Journal of computational physics 1975-01, Vol.18 (4), p.421-439
• Main Authors: Dietrich, D., McDonald, B.E., Warn-Varnas, A.
• Format: Article
• Language: English
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/0021-9991(75)90095-9

A new relaxation method, block-implicit relaxation (BIR), which is applicable to partial difference equations with mesh varying coefficients and irregular boundaries, is compared with the less general Wachspress-optimized ADI method in solving the Poisson-Dirichlet problem on a rectangle. BIR consists of dividing a large computational mesh into several small meshes, and solving the difference equation exactly in each submesh interior. Residuals on the submesh boundaries are reduced by an iterative relaxation scheme. BIR is found superior for all but the largest forcing function scales. The large-scale convergence is accelerated significantly by a least-squares optimization procedure, which requires little additional computation or storage. In application to related sequences of problems for which accurate high-order extrapolation is possible, the new method has the strong advantage of performing such extrapolation with relatively little auxiliary storage or computation. Thus, the new method is well suited for time-implicit time marching models. Application to a time-implicit nonlinear transport equation with diffusion (high Reynolds' number channel flow) is discussed.