A Conservative Semi-Lagrangian Finite Volume Method for Convection-Diffusion Problems on Unstructured Grids

A conservative semi-Lagrangian finite volume method is presented for the numerical solution of convection-diffusion problems on unstructured grids. The new method consists of combining the modified method of characteristics with a cell-centered finite volume discretization in a fractional-step manner where the convection part and the diffusion part are treated separately. The implementation of the proposed semi-Lagrangian finite volume method differs from its Eulerian counterpart in the fact that the present method is applied at each time step along the characteristic curves rather than in the time direction. To ensure conservation of mass at each time step, we adopt the adjusted advection techniques for unstructured triangular grids. The focus is on constructing efficient solvers with large stability regions and fully conservative to solve convection-dominated flow problems. We verify the performance of our semi-Lagrangian finite volume method for a class of advection-diffusion equations with known analytical solutions. We also present numerical results for a transport problem in the Mediterranean sea.