  Reaction-Diffusion equations can present solutions in the form of traveling
waves. Such solutions evolve in different spatial and temporal scales and it is
desired to construct numerical methods that can adopt a spatial refinement at
locations with large gradient solutions. In this work we develop a high order
adaptive mesh method based on Chebyshev polynomials with a multidomain approach
for the traveling wave solutions of reaction-diffusion systems, where the
proposed method uses the non-conforming and non-overlapping spectral
multidomain method with the temporal adaptation of the computational mesh.
Contrary to the existing multidomain spectral methods for reaction-diffusion
equations, the proposed multidomain spectral method solves the given PDEs in
each subdomain locally first and the boundary and interface conditions are
solved in a global manner. In this way, the method can be parallelizable and is
efficient for the large reaction-diffusion system. We show that the proposed
method is stable and provide both the one- and two-dimensional numerical
results that show the efficacy of the proposed method.
