Analysis of a degenerate parabolic cross-diffusion system for ion transport

Finite‐volume scheme for a degenerate cross‐diffusion model motivated from ion transport

“…Assumption (A1) is supposed for simplicity only. Mixed Dirichlet-Neumann boundary conditions can be included in the analysis (see, e.g., [20]), but the proofs become even more technical. Mixed boundary conditions are chosen in the numerical experiments; therefore, the numerical scheme is defined for that case.…”

“…Observe that we employed a double upwinding: one related to the electric potential, defining p u k 0,σ,i , and another one related to the drift part of the flux, V k i,K,σ . The potential is computed via (20) ´λ2…”

“…L 8 bounds and existence of solutions. In order to prove the existence of solutions to ( 15)- (20), we first consider a truncated problem. This means that we truncate the expressions in (18); more precisely, we consider scheme (15), (16), and (20) with…”

“…Our aim is to design a numerical approximation of (1) which preserves the structural properties of the continuous equations. This suggests to use the entropy variables as the unknowns, as it was done in our previous work [20] with simulations in one space dimension. Unfortunately, we have not been able to perform a numerical convergence analysis with these variables.…”

“…If the drift part vanishes, the solution is unique. The existence proof uses a topological degree argument in finite space dimensions, while the uniqueness proof is based on the entropy method of Gajewski [18], recently extended to cross-diffusion systems [20,28]. ‚ Thanks to the "double" upwind structure, the scheme preserves the nonnegativity and upper bound for the concentrations (at least if D i " D for all i).…”