时空耦合谱元方法求解一维Burgers方程

王亚洲; 秦国良; 和文强; 包振忠

doi:10.7652/xjtuxb201701008

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时空耦合谱元方法求解一维Burgers方程

更新时间：2025-07-09

- 时空耦合谱元方法求解一维Burgers方程
- Time-Space Coupled Spectral Element Method for One-Dimensional Burgers Equation
- 西安交通大学学报 2017年51卷第1期页码：45-50
- 作者机构：
  
  西安交通大学能源与动力工程学院,西安,710049
- 作者简介：
- 基金信息：
  
  : 2016-06-01。作者简介: 王亚洲(1993—),男,博士生;秦国良(通信作者),男,教授,博士生导师。基金项目: 国家重点基础研究发展规划资助项目(2012CB026004)
- DOI：10.7652/xjtuxb201701008
  中图分类号： O357
- 网络首发：2017-01-10，
  
  纸质出版：2017
- 稿件说明：
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王亚洲, 秦国良, 和文强, 等. 时空耦合谱元方法求解一维Burgers方程[J]. 西安交通大学学报, 2017,51(1):45-50.

Time-Space Coupled Spectral Element Method for One-Dimensional Burgers Equation[J]. 2017, 51(1): 45-50.
王亚洲, 秦国良, 和文强, 等. 时空耦合谱元方法求解一维Burgers方程[J]. 西安交通大学学报, 2017,51(1):45-50. DOI： 10.7652/xjtuxb201701008.

Time-Space Coupled Spectral Element Method for One-Dimensional Burgers Equation[J]. 2017, 51(1): 45-50. DOI： 10.7652/xjtuxb201701008.

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

To improve the matching of numerical accuracy between time and space for Burgers equation

a time-space coupled spectral element method is proposed to solve one-dimensional Burgers equation. The equation is discretized with the spectral element method in both time and space directions

and detailed derivation is displayed. Additionally

the numerical accuracy and the matching characteristic of time and space of the proposed method are compared with those of three other schemes in which spectral element discretization is adopted in space direction while the backward Euler method

the fourth-order Runge-Kutta method and the fourth-order Adams-Bashforth method are employed

respectively

in time direction. The influences of the numbers of elements and interpolation nodes on the numerical accuracy in time direction are investigated. It is verified that: 1)higher accuracy can be obtained with the proposed time-space coupled spectral element method compared with conventional temporal difference schemes in time direction; the numerical accuracy is continuously promoted with the increasing interpolation order

and the exponential convergence rate is still maintained

which shows a better characteristic of matching between time and space; and 3)increasing the elements or interpolation nodes in time direction with fixed space meshing has slight effect on improvement of the numerical accuracy

which coincides with the related research in literature.

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references

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耿艳辉, 秦国良. Chebyshev谱元法求解含吸收边界的二维均匀稳定流场的声传播 [J]. 西安交通大学学报, 2012, 46(3): 100-106.

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