A Primal-dual Forward-backward Splitting Method for Cross-diffusion Gradient Flows with General Mobility Matrices

TL;DR

This paper introduces a primal-dual forward-backward splitting method to efficiently compute cross-diffusion systems modeled as gradient flows with general mobility matrices, advancing numerical solutions for complex PDEs.

Contribution

The paper develops a novel PDFB splitting algorithm tailored for cross-diffusion gradient flows with matrix mobilities, integrating variational formulations and optimization techniques.

Findings

Demonstrates efficiency on challenging cross-diffusion equations

Provides a new computational approach for gradient flow systems

Validates method through numerical experiments

Abstract

In this work, we construct a primal-dual forward-backward (PDFB) splitting method for computing a class of cross-diffusion systems that can be formulated as gradient flows under transport distances induced by matrix mobilities. By leveraging their gradient flow structure, we use minimizing movements as the variational formulation and compute these cross-diffusion systems by solving the minimizing movements as optimization problems at the fully discrete level. Our strategy to solve the optimization problems is the PDFB splitting method outlined in our previous work \cite{PDFB2024}. The efficiency of the proposed PDFB splitting method is demonstrated on several challenging cross-diffusion equations from the literature.