# A variable nonlinear splitting algorithm for reaction-diffusion systems with self- and cross-diffusion

## TL;DR

This paper introduces a novel nonlinear operator splitting algorithm that efficiently models self- and cross-diffusion effects in reaction-diffusion biological systems, ensuring accuracy and stability through rigorous analysis and numerical validation.

## Contribution

It presents a new splitting method that directly incorporates both self- and cross-diffusion, improving computational efficiency and stability in biological reaction-diffusion models.

## Key findings

- The method is numerically stable under specified criteria.
- The algorithm demonstrates high accuracy in numerical experiments.
- It effectively models complex biological diffusion processes.

## Abstract

Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy.