Multigrid Solution of an Elliptic Boundary Value Problem from Tropical Cyclone Theory

Abstract

We consider the multigrid solution of the transverse circulation equation for a tropical cyclone. First we develop a standard multigrid scheme (SMG) which cycles between different levels of discretization (grids) to efficiently reduce the error in the solution on all scales. Whereas relaxation is inefficient as a solution method, it is used within the multigrid approach as a smoother to reduce the high-wavenumber errors on each grid. The added cost of the coarse grids is small because they contain relatively few points. The efficiency of the SMG scheme is compared to more conventional methods. Gauss-Seidel and successive-over-relaxation (SOR). Results show that the SMG scheme solves to the level of truncation error 26 times faster than an optimal SOR method. In the SMG scheme an unbounded domain is approximated with a wall at a finite radius which leads to significant errors in the numerical solution. To better simulate an unbounded domain, we develop a second scheme (LRMG) which naturally combines local mesh refinement with multigrid processing. In this scheme, the lateral boundary is moved far enough out that the wall boundary condition is realistic, and the grid is coarsened in the outer region so that little additional work is required. Since finer grids are introduced only where needed, the LPMG scheme maintains the usual multigrid efficiency.