时空耦合谱元方法求解一维Burgers方程

发布时间：2018-02-02 04:10

本文关键词： 时空耦合数值精度谱元方法 Burgers方程　出处：《西安交通大学学报》2017年01期 　论文类型：期刊论文

【摘要】：针对Burgers方程在空间离散格式与时间离散格式方面的精度匹配问题,提出了一种时空耦合谱元方法,求解了一维Burgers方程。求解时在时间及空间方向同时采用了谱元方法离散方程,推导了求解过程,比较了空间方向采用谱元离散结合时间方向分别采用向后欧拉方法、四阶Runge-Kutta方法和四阶Adams-Bashforth方法的数值精度以及时空匹配特性,研究了时间方向网格单元数及插值节点数对时空耦合谱元方法数值精度的影响。研究显示:时空耦合谱元方法能够求解Burgers方程且与传统的时间差分方法相比能够获得更高的数值精度;随着空间方向单元内插值阶数的不断增大,时空耦合谱元方法的数值精度不断提高,且保留了指数阶收敛的特点,具有较好的时空匹配特性;当空间网格划分方式固定时,时间方向上增加单元数或单元内插值阶数,对数值精度提高影响不大,这一结论与相关研究结果一致。研究内容对引入与空间谱元方法精度相匹配的时间离散格式具有指导意义。
[Abstract]:A spatiotemporal coupled spectral element method is proposed to solve the accuracy matching problem between spatial discrete schemes and time discrete schemes for Burgers equations. The one-dimensional Burgers equation is solved. The spectral element method is used to discretize the equation in both time and space direction, and the solution process is derived. The spatial direction is compared by spectral element discretization and time direction respectively by backward Euler method. Numerical accuracy and space-time matching characteristics of fourth-order Runge-Kutta method and fourth-order Adams-Bashforth method. The effects of the number of time-oriented grid elements and the number of interpolation nodes on the numerical accuracy of the spatio-temporal coupled spectral element method are studied. The space-time coupled spectral element method can solve the Burgers equation and obtain higher numerical accuracy than the traditional time-difference method. With the increasing of interpolation order in spatial direction element, the numerical precision of spatio-temporal coupled spectral element method is improved continuously, and the characteristic of exponential order convergence is retained, so it has better space-time matching property. When the spatial grid is fixed, the increase of the number of elements or the order of interpolation in the time direction has little effect on the improvement of numerical accuracy. This conclusion is consistent with the related research results. The research content has a guiding significance to introduce the time discrete scheme which matches the precision of the spatial spectral element method.
【作者单位】：西安交通大学能源与动力工程学院;
【基金】：国家重点基础研究发展规划资助项目(2012CB026004)
【分类号】：O241.82
【正文快照】： Burgers方程为二阶非线性偏微分方程,可用于描述湍流、边界层发展、激波等现象,特别是其与Navier-Stokes方程具有类似的形式,因而该方程的求解方法研究具有重要的意义。Kutluay等采用Hopf-Cole变换将Burgers方程转变为线性的导热方程,并通过显式有限差分法进行了求解[1]。Seyd

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