A note on an accelerated high-accuracy multigrid solution of the convection-diffusion equation with high Reynolds number

Abstract

We present a new strategy to accelerate the convergence rate of a high-accuracy multigrid method for the numerical solution of the convection-diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss-Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourthorder compact discretization are investigated numerically.

Integrated fast and high‐accuracy computation of convection diffusion equations using multiscale multigrid method

“…The X-Y line Gauss-Seidel relaxation in lexicographic order performs one sweep of line Gauss-Seidel relaxation along the x-coordinate direction first, then another sweep of the line Gauss-Seidel relaxation along the y-coordinate direction. However, it was shown in [21,25] that merely using the X-Y line Gauss-Seidel relaxation in a standard multigrid method does not provide fast convergence for convection-dominated problems with high Re.…”

“…The actual scaling factor is determined by the absolute values of the convection coefficients at the reference grid point. The detail of how to choose an optimal residual scaling factor with high Re can be found in [21,25]. In our experiment, we tested several scaling factors and only list the numerical results from the best one in Section V. We proposed a multiscale multigrid method in [15] to solve the 2D Poisson equation, which computes the fourth-order solutions on both the fine and coarse grids.…”

Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation

“…From [34, 36, 39], we also know that merely using line relaxation in standard multigrid method still cannot give us the fast convergence for convection dominated 2D problems with relatively large Reynolds number. One simple approach is to properly scale the residual before it is projected to the coarse grid.…”

“…Readers are referred to [11, 39] for more details about how to choose the correct scaling factor. In this paper, we only list the optimal numerical results we got by testing different scaling factors.…”