Higher-order methods for the Poisson equation obtained with geometric multigrid and completed Richardson extrapolation

Abstract

The study presented in this paper consists of a grouping of methods for determining numerical solutions to the Poisson equation (heat diffusion) with high accuracy. We compare the results obtained with classical second-order finite difference method (CDS-2) with fourth-order compact (CCDS-4) and the exponential methods (EXP-4). We accelerate the convergence of the numerical solutions using the geometric multigrid method and then apply the completed Richardson extrapolation (CRE) across the full temperature field. This proposed clustering determined solutions with two orders of accuracy higher for all three methods presented in the study, in addition to recommending the EXP-4 method together with CRE for its accuracy and low computational effort. The evidence for our results was established through qualitative verification, through the assessment of orders of accuracy of the discretization error; and quantitative verification, through the analysis of CPU time and complexity order of the numerical solutions calculated. The numerical solutions of sixth-order of accuracy obtained after proposed CRE methodology using the CCDS-4 and EXP-4 methods are recognized as benchmark solutions for these two classes of methods.