Positivity and boundedness preserving schemes for the fractional reaction-diffusion equation

TL;DR

This paper introduces a semi-implicit numerical scheme for the fractional reaction-diffusion equation that guarantees stability, positivity, and boundedness, with proven convergence rates and application to a predator-prey model.

Contribution

The paper develops a novel semi-implicit scheme for fractional reaction-diffusion equations that preserves positivity and boundedness, with theoretical stability and convergence analysis.

Findings

The scheme is unconditionally stable.

Convergence orders are 1%$2-eta$ in time and 2 in space.

Numerical results confirm positivity and boundedness preservation.

Abstract

In this paper, we design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergent orders are 1 %$2-\alpha$ in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution of the system is positive and bounded. Then we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.