Stability analysis and improvement of the solution reconstruction for cell-centered finite volume methods on unstructured meshes

Abstract

The purpose of this paper is to develop a framework in which one can identify and predict the numerical instability of the steady state solution due to the solution reconstruction for cell-centered finite volume methods on unstructured meshes and to stabilize the problem by optimizing the reconstruction stencil. In this work, we first develop and extend a mathematical method, introduced by Haider and his colleagues, to measure the stability impact of the reconstruction phase for both linear and nonlinear problems regardless of the solution. Second order and third order accurate advection and Burgers problems as well as second order Euler problems are used to present detailed practical results and discussion around the use of the local reconstruction map for stability analysis. This method shows that for a range of different physical problems, increasing the stencil size will usually lead to more stable problems. Additionally, an empirical study is performed which sheds light on connections between the mesh properties and the stability of the reconstruction, which in turn helps choose the reconstruction stencil more wisely. Secondly, we propose a systematic approach to optimize both the shape and the size of the reconstruction stencil for better numerical stability through eigenvalue analysis. In this approach, one can directly optimize the solution reconstruction stencil for every control volume to obtain better numerical stability and convergence properties for steady state problems. A second order accurate Euler problem as well as a third order accurate laminar Navier-Stokes problem are used to showcase the applicability of the algorithm.