Efficient multigrid methods for solving upwind-biased discretizations of the convection equation

Abstract

This paper studies a novel multigrid approach to the solution for a second-order upwind-biased discretization of the convection equation in two dimensions. This approach is based on semi-coarsening and well-balanced explicit correction terms, added to coarse-grid operators to maintain on coarse grids the same cross-characteristic interaction as on the target (fine) grid. Multicolor relaxation schemes are used on all the levels, allowing a very efficient parallel implementation. The results of the smoothing analysis and numerical tests can be summarized as follows: (1) The residual asymptotic convergence rate is about 3 per cycle, which far surpasses the theoretical limit (4/3) predicted for standard multigrid algorithms with full-coarsening. (2) The full multigrid (FMG) algorithm with two V(0,2) cycles on the target grid and just one V(0,2) cycle on all the coarse grids always provides an approximate solution with the algebraic error less than the discretization error. (3) A novel algorithm for deriving a discrete solution approximating the true continuous solution with a relative accuracy given in advance is developed. The computational complexity of this algorithm is (nearly) optimal (comparable with the complexity of the FMG algorithm applied to solve the problem on the optimally spaced target grid).