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Center for Nonlinear Studies |
In this talk we present the development of a family of cell-centered finite volume methods for elliptic problems, focusing on the Poisson equation with heterogeneous and anisotropic diffusion tensors and the steady convection-diffusion equation. The scalar unknowns are associated to the barycenters of the mesh cells. The diffusive flux is numerically approximated by using the gradient formula, which provides a constant vector inside the "diamond" cells built around any mesh edge. The convective flux is approximated by using a piecewise linear representation of the solution inside the control volumes of the mesh. The vertex values that are also needed by this formulation are evaluated as weighted averages of the cell unknowns or as the solution of a second finite volume scheme on a staggered mesh as in the DDFV method. These schemes are second order accurate as shown by numerical experiments and a non-linear version of the numerical diffusive flux ensures the existence of a discrete maximum principle for the numerical solution of the convection diffusion problem.
Host: Konstantin Lipnikov, T-5, x 71719, Mikhail Shashkov. shashkov@lanl.gov, 667-4400