Numerical studies of a class of reaction–diffusion equations with Stefan conditions

ABSTRACT

It is always very difficult to efficiently and accurately solve a system of differential equations coupled with moving free boundaries, while such a system has been widely applied to describe many physical/biological phenomena such as the dynamics of spreading population. The main purpose of this paper is to introduce efficient numerical methods within a general framework for solving such systems with moving free boundaries. The major numerical challenge is to track the moving free boundaries, especially for high spatial dimensions. To overcome this, a front tracking framework coupled with implicit solver is first introduced for the 2D model with radial symmetry. For the general 2D model, a level set approach is employed to more efficiently treat complicated topological changes. The accuracy and order of convergence for the proposed methods are discussed, and the numerical simulations agree well with theoretical results.

Keywords:

• Reaction–diffusion equations
• free boundaries
• level set method
• front tracking method
• diffusive logistic models

2010 Mathematics Subject Classification:

• 65M06

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Science Foundation [DMS1853365] and the Australian Research Council.