Numerical solutions of random mean square Fisher-KPP models with advection

TL;DR

This paper develops a stable numerical scheme for solving random mean square Fisher-KPP models with advection, combining semidiscretization and exponential time differencing, and proves stability in the mean square sense.

Contribution

It introduces a novel two-stage numerical scheme for random Fisher-KPP models with advection, ensuring stability and accuracy in the mean square sense.

Findings

The numerical scheme is stable in the mean square sense.

Comparison with known solutions validates the approach.

The method handles the computational complexity effectively.

Abstract

This paper deals with the construction of numerical stable solutions of random mean square Fisher-KPP models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear unhomogeneous system of random differential equations. Then, by extending to the random framework the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity the results are illustrated by comparing the results with a test problem where the exact solution is known.