Exponential Time Differencing Method for a Reaction- Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries, the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes, while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary. It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties. Applying an implicit scheme may be able to remove the stability constraints on the time step, however, it usually requires solving a large global system of nonlinear equations for each time step, and the computational cost could be significant. Integration factor (IF) or exponential time differencing (ETD) methods are one of the popular methods for temporal partial differential equations (PDEs) among many other methods. In our paper, we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries. In particular, we rewrite all ETD schemes into a linear combination of specific F-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications, which offers significant computational advantages with adaptive Krylov subspaces. In addition, we extend this method by incorporating the level set method to solve the free boundary problem. The accuracy, stability, and efficiency of the developed method are demonstrated by numerical examples.