一类带参数的非线性奇异摄动问题的自适应移动网格算法

An Adaptive Moving Grid Algorithm for a Class of Nonlinear Singularly Perturbed Parameterized Problems

本文讨论一类带参数的非线性奇异摄动问题的自适应移动网格方法.首先,在任意非均匀网格下,利用向后欧拉公式对方程进行离散,并给出相应的局部截断误差.然后,基于局部截断误差和网格等分布原理,利用精确解的弧长函数,证明半离散格式下自适应移动网格算法是一阶收敛的.同时,基于近似的弧长控制函数,给出易于实现的网格生成算法,并给出全离散格式下的后验误差估计.最后,数值实验结果验证了本文所给出的理论结果.

An adaptive moving grid algorithm is discussed to solve a class of nonlinear singularly perturbed parameterized problems.At first,the backward Euler method is used to discretize the presented problems on an arbitrary nonuniform mesh,and give the corresponding local truncation error.Then,based on this local truncation error and mesh distribution principle,we prove that the semi-discretized adaptive moving grid method is first-order convergence by using the exact arc-length monitor function.Meanwhile,based on the approximation arc-length monitor function,a mesh generation algorithm which is easy to implement is given,and an a posteriori error estimation is derived for the fully-discretized scheme.At last,numerical results are given to illustrate the theoretical results.