engleski

THE EIGENSPECTRUM OF A FINITE VOLUME MATRIX

The discretisation of partial differential equations using the Finite Volume Method results in linear systems with square, sparse coefficient matrices, which have symmetric addressing. The dimension of the coefficient matrix is equal to the number of finite volumes in the spatial discretisation, i.e. computational mesh. The sparseness pattern of the coefficient matrix is determined by the position of the elements inside the matrix and can affect the performance of iterative linear solvers [1]. The convergence of these iterative methods depends greatly on the eignespectrum of the iteration matrix, which is often correlated with the eigenvalues of the coefficient matrix. In this paper, we shall investigate the eigenspectrum, i.e. eigenvalues and eigenvectors of coefficient matrices obtained from the Finite Volume discretization of partial differential transport equations. We shall compare the effects of mesh properties: number of cells, mesh connectivity (structured, unstructured), cell type and size, cell anisotropy and cell volume ratio. The tests will be conducted for cases with single-phase, incompressible, laminar and turbulent flows. The eigenvalues shall be calculated for velocity and pressure coefficient matrices and compared for cases with dominant convection and diffusion transport. We shall implement and employ the power method [2] for calculating the dominant eigenvalues, as well as an external library, coupled to OpenFOAM.

eigenspectrum, iterative linear algorithms, finite volume discretisation

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