Abstract
The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches.
Highlights
The reaction-diffusion equations over a changing domain to describe the dynamics of a two-species competition-diffusion model usually take the following form
The system of reaction-diffusion equations with moving boundaries has been intensively studied analytically in recent years, very little numerical work has been done in this field due to numerical challenges in tracking free boundaries
The level set method is very robust in handling topological changes, sometimes it is very hard to achieve high order accuracy, especially near the fronts
Summary
The reaction-diffusion equations over a changing domain to describe the dynamics of a two-species competition-diffusion model usually take the following form,. In the rest of this paper, we will take two-species Lotka-Volterra type competition functions as an example to demonstrate the numerical methods. To efficiently handle the moving boundaries, level set methods [19,20,21,22,23,24] and front tracking methods [25,26,27,28] are two popular numerical approaches. One distinct feature of front tracking [29,30,31,32,33,34] is using a pure Lagrangian approach to explicitly track locations of interfaces, but it is difficult to handle topological bifurcations in high dimensions with interaction of two free boundaries, while the level set method can efficiently overcome such difficulties.
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