An application of the Multi-Grid method to the construction of initial data for Brill Waves

Abstract

A linear constraint equation arising from the 3+1 description of general relativity has been solved using both a standard Successive-Over-Relaxation (SOR) scheme and a Multi-Grid (MG) algorithm. In a comparison of computer timings we show that MG can reach a specified convergence criterion nearly 10 times faster than SOR for grids of ∼ 64 × 64 zones, where this factor increases with grid size. In addition it is shown that iterating to a specified convergence criterion can result in wasted work since the truncation error of the solution can be reached well before such a criterion is satisfied. Since MG provides estimates of the solution errors we can assess the minimal amoun of work required to solve the problem. Then it is found that our equation can be solved on a 256 × 256 zones grid using MG in approximately the same time as SOR would need for a ∼ 25 × 25 zones grid if a usual convergence criterion were employed.