Abstract:

For solving singular perturbed interior layer problems, a numerical method is proposed to estimate the interior layer width. First, by using the domain decomposition method, the original problem is divided into two parts：the reduced problem and the interior problem. Both problems can be discretized and solved by the rational spectral collocation method in barycentric form. In order to determine the interior layer width, it designs a nonlinear optimization problem based on the continuity condition of first order derivatives. Further, a class of nonlinear interior layer problems is considered and Adomian decomposition method is introduced to deal with the nonlinear term. Numerical results illustrate the feasibility and validity of the proposed numerical method for both linear and nonlinear problems, good accuracy can be obtained when the interior layer width is estimated effectively.

Key words: singular perturbed, interior layer, width estimation, rational spectral collocation method, domain decomposition

摘要：

针对奇异摄动内层问题的数值计算，提出了一种内层宽度的估计方法。通过区域分解将原问题拆分为退化问题和内层问题两部分，采用重心形式的有理谱配点法进行离散求解。为了确定内层的宽度，设计了基于一阶导数连续性条件的非线性优化问题。进一步，借助Adomian分解法考虑了一类非线性内层问题的线性化求解。数值实验验证了该方法的可行性和有效性，对线性和非线性问题均给出了内层宽度的一个合理估计，同时数值解也达到了不错的计算精度。

关键词: 奇异摄动, 内层, 宽度估计, 有理谱配点法, 区域分解