Darcy's Law with a Source term

TL;DR

This paper develops a new numerical scheme for Darcy's law with a pressure-dependent source, maintaining key properties and proving convergence to solutions of complex PDE systems like tumor growth models.

Contribution

It introduces a novel variant of the JKO scheme that handles mass changes and source terms while preserving important analytical properties.

Findings

The scheme satisfies a comparison principle.

The scheme exhibits uniform $L^1$-equicontinuity.

Convergence to PDE solutions including tumor growth models is proven.

Abstract

We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform $L^1$-equicontinuity. Many of these properties are new even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.