Fast stable finite difference schemes for nonlinear cross-diffusion

TL;DR

This paper introduces two operator splitting schemes for nonlinear cross-diffusion models that significantly reduce computational complexity and enable real-time applications by ensuring stability and efficient matrix factorization.

Contribution

The paper presents novel stable operator splitting schemes for nonlinear cross-diffusion, improving computational efficiency and enabling real-time processing.

Findings

Proposed two operator splitting schemes with proven stability.

Achieved stable factorization of system matrix for fast computation.

Demonstrated potential for real-time cross-diffusion simulations.

Abstract

The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete $L^2$ energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for \textit{on-the-fly} applications.