Sinc配点法与sinh变换求解奇异摄动过渡层问题  

Sinc collocation method and sinh transform for solving singular perturbed problems with a transition layer

作　　者：邵文婷[1] 郑烁宇 SHAO Wenting;ZHENG Shuoyu(School of Mathematics,Physics and Statistics,Shanghai Polytechnic University,Shanghai 201209,China;School of Mathematical Sciences,Tongji University,Shanghai 200092,China)

机构地区：[1]上海第二工业大学数理与统计学院,上海201209 [2]同济大学数学科学学院,上海200092

出　　处：《信阳师范大学学报(自然科学版)》2025年第1期92-100,共9页Journal of Xinyang Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(12001362,11526132);上海市自然科学基金项目(16ZR1412700);上海市教委高校优秀青年教师专项基金项目(ZZEDG15008)。

摘　　要：针对一类奇异摄动过渡层问题,采用具有高精度的Sinc配点法进行离散求解。利用双指数变换,将自然定义在无界区域上的Sinc配点法推广到求解带有齐次Dirichlet边界条件的两点边值问题。对于非齐次Dirichlet边值问题,设计线性函数对未知函数进行改造,将原问题转化为齐次Dirichlet边值问题。对于过渡层的处理,利用sinh变换将Sinc节点在奇性处进行加密,实现较少的节点即可达到较高的精度。基于双指数变换,提出了有界区域上重心形式的Sinc插值公式来改善非插值节点上的近似效果。数值实验验证了结合双指数变换和sinh变换的Sinc配点法在求解过渡层问题、具有激波的非定常Burgers’方程上的高效性。High-order Sinc collocation method was used to solve a class of singular perturbed problems with a transition layer.Combined with the double exponential transform,Sinc collocation method defined naturally on the unbounded domain as available to the two-point boundary value problem with homogenous Dirichlet boundary conditions.For non-homogenous Dirichlet boundary conditions,a linear function was designed to reconstruct the unknown function.That is,it transformed the original problem into the problem with homogenous Dirichlet boundary conditions.The sinh transform was utilized to make the Sinc nodes dense in the location of the transition layer,which leaded to good accuracy with less computational cost.Based on the double exponential transform,a Sinc-barycentric form interpolation formulae defined on the bounded domain was proposed to improve the performance of the numerical approximation of non-node function values.Numerical results verified the computational efficiency of the Sinc collocation method,which combined with the double exponential transform and the sinh transform,for solving singular perturbed problems with a transition layer and the non-stationary Burgers’equation with a shock wave.

关 键 词：过渡层 Sinc配点法 双指数变换 sinh变换 重心形式Sinc插值公式 

分 类 号：O241.81[理学—计算数学]