Structure-Preserving Numerical Schemes for a Time-Fractional Porous Medium Equation with Nonlocal Pressure

Abstract

In this work, we study numerical approximations of the equation \begin{equation} \partial_t^αu-\nabla\cdot\left(u^{m-1}\nabla(-Δ)^{-σ}u\right)=0, \end{equation} where $0<α<1$, $0<σ<1$ and $m>1$. The Caputo derivative is discretized by the L1 formula, while the nonlinear nonlocal flux is approximated by a conservative upwind finite-difference scheme. We establish several structural properties of the implicit scheme in arbitrary space dimension, including a discrete maximum principle, non-negativity, mass conservation, $\ell^\infty$ and $\ell^2$ stability, existence of non-negative discrete solutions, and a discrete energy-dissipation estimate. Our assumptions include natural symmetry, a sign condition on the sum of the directional coefficients, and decay on the discrete weights. We also derive a consistency estimate for the complete nonlinear spatial operator. For rigorous convergence analysis, we restrict ourselves to one space dimension and introduce the cumulative variable associated with the density. This transforms the discrete problem into a monotone approximation of an equation involving the fractional Laplacian. Interestingly, for semigroup-based pressure-gradient weights, the resulting discrete operator is identified with the fractional discrete Laplacian. Next, using comparison and contraction properties of the integrated scheme, for $m\geq2$ and under the stated spatial regularity and weighted temporal consistency assumptions, we prove uniform convergence of the cumulative numerical solution and convergence of the corresponding density against compactly supported smooth test functions. The temporal estimates allow a weak singularity at the initial time. We also analyze an explicit variant of the scheme and derive a sufficient CFL condition covering the full range $0<σ<1$.