Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations
For researchers in numerical analysis and computational fluid dynamics, this work provides a scheme that maintains a crucial physical property on challenging meshes, but it is an incremental improvement over existing DDFV methods.
The authors propose a nonlinear Discrete Duality Finite Volume scheme for drift-diffusion equations that preserves the energy/energy dissipation relation on distorted meshes, ensuring accurate long-time behavior. They prove existence of positive solutions, convergence to weak solutions, and provide numerical evidence of good behavior as discretization parameters tend to zero and time goes to infinity.
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.