Red-black Relaxation Research Articles

A space-time multigrid method for poroelasticity equations with random hydraulic conductivity

In this work, we propose a space-time multigrid method for solving a finite difference discretization of the linear Biot’s model in two space dimensions considering random hydraulic conductivity. This method is an extension of the one developed by Franco et al. [1] and which was applied to the heat equation. Particularly for the poroelasticity problem, we need to use the Tri-Diagonal Matrix Algorithm (TDMA) for systems of equations with multiple variables (block TDMA) solver on the planes xt and yt with zebra-type wise-manner (the domain is divided into planes and solve each odd plane first and then each even plane independently [2]). On the planes xy, the F-cycle multigrid with fixed-stress smoother is improved by red-black relaxation. A fixed-stress smoother is compound solver, where the mechanical and flow parts are addressed by two distinctive solvers and symmetric Gauss-Seidel iteration. This is the basis of the proposed method which does not converge only with the standard line-in-time red-black solver [1]. This combination allows the code parallelization. The use of standard coarsening associated with line or plane smoothers in the strong coupling direction allows an application in real world problems and, in particular, the random hydraulic conductivity case. This would not be possible using the standard space-time method with semicoarsening in the strong coupling direction. The spatial and temporal approximations of the problem are performed, respectively, by using Central Difference Scheme (CDS) and implicit Euler methods. The robustness and excellent performance of the multigrid method, even with random hydraulic conductivity, are illustrated with numerical experiments through the average convergence factor.

Multigrid smoothing for symmetric nine-point stencils

Analytical formulae are obtained for the smoothing factors yielded by damped Jacobi relaxation and by red-black relaxation applied to symmetric nine-point stencil discretizations of elliptic partial differential operators in 2D. The results include point and line relaxation and full and partial coarsening. Several unusual results are implied by the formulae. Numerical results of multigrid cycles match the analytical predictions well.

Ordering Effects on Relaxation Methods Applied to the Discrete One-Dimensional Convection-Diffusion Equation

The authors present an analysis of relaxation methods for the one-dimensional discrete convection-diffusion equation based on norms of the iteration matrices. In contrast to standard analytic techniques that use spectral radii, these results show how the performance of iterative solvers is affected by directions of flow associated with the underlying operator, and by orderings of the discrete grid points. In particular, for problems of size n, relaxation against the flow incurs a latency of approximately n steps in which convergence is slow, and red-black relaxation incurs a latency of approximately ${n / 2}$ steps. There is no latency associated with relaxation that follows the flow. These results are largely independent of the choice of discretization.