  For reaction-diffusion equations in irregular domain with moving boundaries,
the numerical stability constraints from the reaction and diffusion terms often
require very restricted time step size, while complex geometries may lead to
difficulties in 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 differencing time (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 {\phi}-functions and apply one start-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.
