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

Matthew A. Beauregard, Joshua L. Padgett

TL;DR

The paper develops an adaptive nonlinear operator splitting scheme for reaction-diffusion systems with self- and cross-diffusion, combining Crank-Nicolson time stepping with an ADI-type splitting to decouple multidimensional diffusion operators. It proves second-order accuracy in time and space and establishes a CFL-type stability condition using the matrix logarithmic norm, while relaxing regularity requirements and accommodating general boundary conditions. Numerical experiments with Dirichlet and Neumann BCs validate the convergence rate, demonstrate computational efficiency with $O(N^2)$-like cost per step, and illustrate both global existence under certain parameter regimes and finite-time blow-up under others. The approach is poised for extension to more complex systems and scalable HPC implementations, offering a robust tool for simulating nonlinear diffusion-driven ecological dynamics.

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.

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

TL;DR

The paper develops an adaptive nonlinear operator splitting scheme for reaction-diffusion systems with self- and cross-diffusion, combining Crank-Nicolson time stepping with an ADI-type splitting to decouple multidimensional diffusion operators. It proves second-order accuracy in time and space and establishes a CFL-type stability condition using the matrix logarithmic norm, while relaxing regularity requirements and accommodating general boundary conditions. Numerical experiments with Dirichlet and Neumann BCs validate the convergence rate, demonstrate computational efficiency with -like cost per step, and illustrate both global existence under certain parameter regimes and finite-time blow-up under others. The approach is poised for extension to more complex systems and scalable HPC implementations, offering a robust tool for simulating nonlinear diffusion-driven ecological dynamics.

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.

Paper Structure

This paper contains 9 sections, 11 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Splitting algorithm with cross-diffusion only
  3. Splitting algorithm with self and cross-diffusion
  4. Stability and Convergence
  5. Numerical Experiments
  6. Example 1.
  7. Example 2.
  8. Example 3.
  9. Conclusions

Key Result

Theorem 2.1

The induced error between the factorizations and Eqs. (crank1)-(crank2), respectively, is $\mathcal{O}(\tau_k^3)$.

Figures (2)

  • Figure 1: A log-log plot of the computational time, in seconds, versus $N$ after $1000$ iterations. The temporal step is held constant, $\tau = 10^{-6}$, while $\delta=1/(N-1)$. A linear least squares approximates the slope of the line to be $1.75681$. This indicates that the computational time is proportional to $N^{1.654628}$. Since this is slower than $N^2$, then the proposed nonlinear splitting scheme is highly efficient.
  • Figure 2: A panel showing the (a) initial condition, (b) population at $t=.5$ and (c) just prior to finite time blow-up at $t=.7945$, from left to right, respectively.

Theorems & Definitions (23)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • Definition 4.1
  • Remark 4.1
  • ...and 13 more