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.
翻译:在这项工作中,我们开发了一种基于Chebyshev 聚积法的高排序适应性网格方法,其基础是Chebyshev 聚积法,并采用多域法,用于反应扩散系统的流动波解方法,其中拟议方法使用不兼容和不重叠的谱谱谱多域法,对计算网段进行时间调整。与现有的反应扩散方程式多多域光谱法相反,拟议的多域谱法首先解决了每个子域的指定PDE,而边界和界面条件则以全球方式解决。这样,该方法可以平行使用,对大型反应扩散系统有效。我们表明,拟议方法是稳定的,提供了显示拟议方法功效的一维和二维数值结果。