带插值的自适应网格重构算法求解带移动热源的反应扩散方程的理论分析

Analysis of an adaptive remeshing algorithm with interpolations for reactiondiffusion equations with traveling heat source

本文研究一个带插值的网格重构算法求解一类带移动热源的反应扩散方程.算法包括两步:第一步是用旧时间网层上的计算解计算新时间层上的空间网格;第二步是使用有限差分方法在新时间层空间网格上离散方程,并且将旧时间层上计算解的插值作为初始值.对于时间,我们获得了一阶收敛结果.对于空间,我们证明了使用线性插值算法的一阶收敛性和使用二次插值算法的二阶收敛性.数值例子肯定了本文的理论结果.

This paper studies an adaptive remeshing algorithm with interpolations for solving a kind of reactiondiffusion equations with moving source terms. The algorithm has two steps： The first step is to compute the spatial meshes at the new time level using the computational solutions at the old time level; The second step is to discretize the equations using finite difference methods on the spatial meshes at the new time level with the interpolations of solutions at the old time level as the initial values. For the time, a first-order convergence is obtained. For the space, the first-order convergence is proved for the algorithm with linear interpolations, and second-order convergence with quadratic interpolations. Numerical examples are carried out to confirm the theoretical findings.