摘要
本文通过引入一种新的增广变换,发展了改进的偏牛顿校正算法,建立并证明了一类四阶非线性偏微分方程边值问题的新解与该问题零核空间的密切关系,去掉了标准收敛假设,使证明更简洁明了.分情况验证了该方程满足在Nehari子流形上全局分离定理的条件,该分离定理为本文算法成功找到新解提供理论保障.提出了二维非线性四阶偏微分方程Dirichlet边值问题的插值投影Legendre-Galerkin谱方法,通过构造插值算子和投影算子,对线性算子以及非线性项的处理进行了优化,得到原问题的代数方程,通过验证,其与经典谱方法具有相同的条件数并都达到谱精度.实验结果表明,此方法与经典的谱方法或拟谱方法具有相同的收敛阶,但计算所需CPU时间更少,且能计算出更多的解.
In this paper,we introduce a new augmented transformation,develop an improved partial Newton-correction method,establish and prove the close relationship between the new solutions of the boundary value problem for the fourth order nonlinear partial differential equation and their zero kernel space,remove the standard convergence assumption and make the proof more concise.We prove that our problem satisfies the conditions of the global separation theorem on Nehari manifold in different cases.The global separation theorem provides theoretical guarantee for our algorithm to find new solutions successfully.The Legendre Galerkin spectral method with interpolation projection is proposed for boundary value problems of two-dimensional nonlinear fourth order partial differential equations.By constructing interpolation operators and projection operators,the linear operators and nonlinear terms are optimized,and the algebraic equation of the original problem is obtained.It has the same condition number as the classical spectral method and achieves the same spectral accuracy.Finally,some computing results show that our method has the same high order of convergence as the classical spectral method or pseudo-spectral method,and requires less CPU time,can compute as many solutions as possible.
作者
王旭浩
王培培
李昭祥
陈先进
Wang Xuhao;Wang Peipei;Li Zhaoxiang;Chen Xianjin(Department of Mathematical Sciences,Shanghai Normal University,Shanghai 200234,China;School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China)
出处
《数值计算与计算机应用》
2023年第3期285-304,共20页
Journal on Numerical Methods and Computer Applications
基金
国家自然科学基金(11871043,12271366,12171322)
上海市自然科学基金(21ZR1447200,22ZR1445500)
上海市科技计划(20JC1414200)资助。
关键词
四阶非线性偏微分方程
改进的偏牛顿校正方法
插值投影Legendre-Galerkin谱方法
多解
Fourth order nonlinear partial differential equation
Improved partial Newton-correction method
The Legendre Galerkin spectral method with interpolation projection
Multiple solutions
