Numerical analysis of a nonlinear free-energy diminishing discrete duality finite volume scheme for convection diffusion equations. (English) Zbl 1401.65096
Summary: We propose a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy/energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. We establish the existence of positive solutions to the scheme. Based on compactness arguments, the convergence of the approximate solution towards a weak solution is established. Finally, we provide numerical evidences of the good behavior of the scheme when the discretization parameters tend to 0 and when time goes to infinity.
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
convection diffusion equation; discrete duality finite volumes; convergence analysis; discrete entropy methodReferences:
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