Performance of Geometric Multigrid Method forTwo-Dimensional Burgers’ Equations with Non-Orthogonal, Structured Curvilinear Grids

Abstract

This paper seeks to develop an efficient multigrid algorithm for solving the Burgers problem with the use of non-orthogonal structured curvilinear grids in L-shaped geometry. For this, the differential equations were discretized by Finite Volume Method (FVM) with second-order approximation scheme and deferred correction. Moreover, the algebraic method and the differential method were used to generate the non-orthogonal structured curvilinear grids. Furthermore, the influence of some parameters of geometric multigrid method, as well as lexicographical Gauss–Seidel (Lex-GS), η-line Gauss–Seidel (η-line-GS), Modified Strongly Implicit (MSI) and modified incomplete LU decomposition (MILU) solvers on the Central Processing Unit (CPU) time was investigated. Therefore, several parameters of multigrid method and solvers were tested for the problem, with the use of nonorthogonal structured curvilinear grids and multigrid method, resulting in an algorithm with the combination that achieved the best results and CPU time. The geometric multigrid method with Full Approximation Scheme (FAS), V-cycle and standard coarsening ratio for this problem were utilized. This article shows how to calculate the coordinates transformation metrics in the coarser grids. Results show that the MSI and MILU solvers are the most efficient. Moreover, the MSI solver is faster than MILU for both grids generators; and the solutions are more accurate for the Burgers problem with grids generated using elliptic equations.