Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers
Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers
复制标题
具有移动边界层的抛物型方程有限差分格式ε-一致收敛的必要条件
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发表时间：
2007
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通讯作者：
G. Shishkin
中科院分区：
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作者：
G. Shishkin
研究了一类奇摄动抛物型反应扩散方程边值问题的网格逼近，其中边界沿x轴沿着正方向移动.对于参数λ（这是方程的高阶导数的系数，λ ∈（0，1]）的小值，移动边界层出现在左侧边界S1 L的邻域中。在固定边界层的情况下，在层中凝聚的分段均匀网格上的经典有限差分格式以O（N− 1 lnN + N 0）的速度收敛，其中N和N 0定义了x和t中的网格点数。对于本文所研究的问题，基于均匀网格的经典有限差分格式仅在条件N−1 + N 0 −1 ≠ 0时收敛。结果表明，在关于x和t凝聚在S1 L邻域内的矩形网格上的差分格式类中，在条件N−1 + N 0 −1 ≤ N − 1/2下不能达到收敛。检查的宽度是相似的Kolmogorov的宽度使得有可能建立必要和充分的条件的一致收敛的近似的解决方案的边界值问题。这些条件被用来设计一个以O（N− 1 lnN + N 0）的速度收敛的方案。
A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S1L. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ɛ-uniformly at a rate of O(N−1lnN + N0), where N and N0 define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N−1 + N0−1 ≪ ɛ. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S1L with respect to x and t, the convergence under the condition N−1 + N0−1 ≤ ɛ1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ɛ-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ɛ-uniformly at a rate of O(N−1lnN + N0).