A case-study in open-source CFD code verification, Part I: Convergence rate loss diagnosis

This study analyzes the influence of cell geometry on the numerical accuracy of convection-diffusion operators in OpenFOAM. The large variety of solvers and boundary conditions in this tool, as well as the precision of the finite-volume method in terms of mesh quality, call for a verification process performed in steps. The work is divided into two parts. In the first (the current manuscript), we focus on the diffusion operator, which has been found to exhibit a loss in convergence rate. Although the cell-centered finite volume approach underlying OpenFOAM should preserve a theoretical second order convergence rate, loss of convergence order is observed when non-orthogonal meshes are used at the boundaries. To investigate the origins of this problem, the method of manufactured solutions is applied to yield analytical solutions for the Poisson equation and compute the numerical error. The root cause is identified and corrections to recover second-order convergence are proposed. In part two of this investigation, we show how convergence can be improved, and present results for problems described by the Poisson and Navier-Stokes equations.