| The propagation of spiral waves often refers to the no-flux boundary condition. There are many models simulating realistic systems, whose boundaries are irregular. How to construct and solve these irregular no-flux boundaries exactly is important to the simulation and the research of various realistic systems.We can use finite difference method to overcome this difficulty by defining additional external grid points. However, the undesirable feature of this approach is that it is possible for the same grid cell to have different values depending on which neighboring cell is being updated.Another commonly used technique, finite element method, is able to handle no-flux boundary condition naturally. However, these methods are generally slower than finite differences for equivalent grid spacing and are more cumbersome to implement.We introduce a new method in the chapter 2 which can implement no-flux boundary condition in arbitrary geometries. The algorithm called phase field method has been applied successfully to a wide range of problems including dendritic solidification, viscous fingering, crack propagation, the tumbling of vesicles and intracellular dynamics. This method has the chief advantage that it avoids the need to track the interface explicitly to establishing no-flux boundary condition by introducing an auxiliary field that makes the interface spatially diffuse. What's more, phase field method can be extended to modeling moving boundaries.In chapter 3 and chapter 4, we researched the relationship between the accuracy of phase field method and the curvature of boundary by simulating the Barkley model and the FHN model. When the curvature is 0, we found that the difference between the phase field method and the finite difference method is very small. When the curvature is not 0, the difference is concerned to the change of curvature. Thus, we can say there is a close correlation between the accuracy of phase field method and the boundary curvature.
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