Darcy's Law with a Source term
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 -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.
Cite
@article{arxiv.2006.09558,
title = {Darcy's Law with a Source term},
author = {Matt Jacobs and Inwon Kim and Jiajun Tong},
journal= {arXiv preprint arXiv:2006.09558},
year = {2020}
}