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UBC Theses and Dissertations

Numerical stability analyses and improvement of cell-centered finite volume methods on unstructured meshes Zangeneh, Reza


The purpose of this thesis is to develop a framework in which one can detect and automatically improve the numerical stability of cell-centered finite-volume calculations on unstructured meshes through optimization schemes that modify the mesh, the solution reconstruction, or the boundary conditions. In this process, eigenanalysis and the gradients of the eigenvalues with respect to different parameters are used to ensure energy stability of the system, consequently resulting in convergence. First, gradients of eigenvalues with respect to the local changes in the mesh are calculated to find directions and magnitudes of mesh movements that will make the Jacobian of a semi-discrete system of equations negative semi-definite. These mesh movement vectors are used to modify the mesh locally. The numerical results show that the proposed methods are able to locate the problematic parts of the mesh responsible for instabilities for several physical problems and to correct the instability. Secondly, I develop a mathematical method, introduced by Haider et al., to measure the stability impact of the reconstruction for high order and nonlinear problems, regardless of the solution. Second order and third order accurate advection and Burgers and 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 increasing the stencil size will lead to more stable problems. An empirical study is performed which sheds light on connections between the mesh properties and the stability of the reconstruction. I also 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. The rightmost eigenpairs of the spatially discretized system of equations are used to obtain an optimized boundary condition which will ensure the energy stability of the system. Lastly, the sensitivity of the rightmost eigenvalues to the solution is measured to investigate the effect of using surrogate solutions for the purpose of linearizing the semi-discretized Jacobian.

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