# A boundary preserving numerical scheme for the Wright-Fisher model

## TL;DR

This paper introduces a new explicit numerical scheme for non-linear stochastic differential equations that preserves boundary conditions, with proven strong convergence and extensions to multidimensional cases, applicable in biological and physiological modeling.

## Contribution

It presents a boundary-preserving explicit numerical scheme for non-linear SDEs, extending previous semi-discrete methods and proving strong convergence for models in population and cellular dynamics.

## Key findings

- The scheme is strongly convergent for the class of SDEs considered.
- The method preserves boundary conditions in numerical simulations.
- Extension to multidimensional SDEs demonstrated.

## Abstract

We are interested in the numerical approximation of non-linear stochastic differential equations (SDEs) with solution in a certain domain. Our goal is to construct explicit numerical schemes that preserve that structure. We generalize the semi-discrete method \emph{Halidias N. and Stamatiou I.S. (2016), On the numerical solution of some non-linear stochastic differential equations using the Semi-Discrete method, Computational Methods in Applied Mathematics,16(1)} and propose a numerical scheme, for which we prove a strong convergence result, to a class of SDEs that appears in population dynamics and ion channel dynamics within cardiac and neuronal cells. We furthermore extend our scheme to a multidimensional case.