Espen Jakobsen, NTNU – Norwegian University of Science and Technology
We carefully construct and prove convergence of the first numerical discretisation of the porous medium equation with fractional pressure. This model was introduced by Caffarelli and Vazques in 2011, and is currently one of two main nonlocal extensions of the local porous medium equation. It has finite speed of propagation and comes from a nonlocal Fick’s law, but as opposed to the other extension, it does not satisfy the comparison principle. Without comparison, the analysis is difficult. Uniqueness is only known in 1d, where one can exploit that the integral of the solution (the “cummulative density”) satisfies an “integrated equation”, a nonlocal quasilinear parabolic equation in nondivergence form that can be analysed through viscosity solution methods. Our numerical method then loosely speaking consists in discretising this “integrated” equation with a difference quadrature scheme and then compute the original solution via numerical differentiation. Using upwinding in a untraditional way, we obtain a new type of monotone schemes that allows for convergence analysis via the Barles-Perthame-Souganidis half-relaxed limit method. Combining this result with tightness arguments, we then prove convergence of the approximations of the original problem in the Rubinstein-Kantorovich (bounded Lipschitz) distance uniformly in time. Our results cover both absolutely continuous and Dirac/point mass initial data, and in the latter case, machinery for discontinuous viscosity solutions are needed in the analysis.
It turns out that our scheme can be seen as a conservative upwind finite volume discretisation of the original problem. This scheme is somewhat similar to recent schemes for aggregation equations preserving the gradient flow structure of the problem. This work is joint with Felix del Teso, UAM, Madrid.
F. del Teso and E. R. Jakobsen A convergent finite difference-quadrature scheme for the porous medium equation with nonlocal pressure. Preprint: arxiv:2303.05168