A Chebyshev multidomain adaptive mesh method for reaction-diffusion equations

Reaction-diffusion equations can present solutions in the form of traveling waves with stiff wave fronts. Such solutions evolve in different spatial and temporal scales, and it is desired to construct numerical methods that can adopt spatial refinement at locations where the solution becomes stiff. In this work, we develop a high-order adaptive mesh method based on Chebyshev spectral methods with a multidomain approach for the traveling wave solutions of reaction-diffusion equations. The proposed method uses the non-conforming and non-overlapping spectral multidomain method with the temporal adaptation of the computational mesh. The spectral multidomain methods have been used for solving PDEs including the reaction-diffusion equations. However, the non-conformal, non-overlapping and adaptive mesh spectral methods have not been used in the reaction-diffusion community. The proposed multidomain spectral method solves the given reaction-diffusion equations in each subdomain locally first and the boundary and interface conditions are enforced in a global manner. In this way, the method can be parallelizable and is efficient for large reaction-diffusion systems. We show that the proposed method is stable and accurate for solving reaction-diffusion equations with stiff traveling wave solutions. We provide both one- and two-dimensional numerical results that show the efficacy of the proposed method. The application of the adaptive multidomain spectral methods to the reaction-diffusion equations, that yield stiff traveling wave solutions, is new and needs further investigation.