# Numerical modelling of convection-diffusion-reaction problems with free boundary in 1D

## TL;DR

This paper introduces a numerical method for solving 1D convection-diffusion-reaction problems with free boundaries, transforming the problem into a fixed domain and modeling interface evolution via ODEs, applicable to various free boundary problems.

## Contribution

It presents a novel numerical approach that models interface evolution using local wave shapes and transforms free boundary problems into fixed domain problems for efficient computation.

## Key findings

- The method accurately approximates problems with finite interface speed.
- It can handle some problems with infinite interface speed.
- Numerical results agree well with analytical solutions.

## Abstract

We discuss a numerical method for convection-diffusion-reaction problems with a free boundary in 1D. The method is based on the numerical modelling of the interface evolution, the transformation to a fixed domain problem and the approximation by an ODE system. The interface evolution is modelled by means of the local shape of the corresponding travelling wave solution. The method can be applied to many free boundary problems with a finite speed of the interface. The presented method can also approximate some problems with an infinite speed of the interface for damped travelling wave type solutions. In the numerical experiments we compare our numerical solution with the analytical ones for some problems.