IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Numerical methods for a class of reaction-diffusion equations with free boundaries

  • Feb. 8, 2019
  • 2:30 p.m.
  • LeConte 317R

Abstract

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of the spreading population. There are several numerical difficulties to efficiently handle such systems. On 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. To overcome these difficulties, we first transform the one-dimensional problem with moving boundaries into a system with a fixed computational domain and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems for the 1D model with moving boundary. We also introduce a front tracking method coupled with an implicit solver for the 2D model with radial symmetry. 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.

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