Spectral Element Method for Convection-Diffusion Equation with Stability Analysis

doi:10.7652/xjtuxb201501001

您当前的位置：

首页 >

文章列表页 >

Spectral Element Method for Convection-Diffusion Equation with Stability Analysis

更新时间：2025-07-09

- Spectral Element Method for Convection-Diffusion Equation with Stability Analysis
- Vol. 49, Issue 1, Pages: 1-6(2015)
- 作者机构：
  
  西安交通大学流体机械研究所,西安,710049
- 作者简介：
- 基金信息：
- DOI：10.7652/xjtuxb201501001
  CLC： O357.1
- Online First：10 January 2015，
  
  Published：2015
- 稿件说明：
移动端阅览
Spectral Element Method for Convection-Diffusion Equation with Stability Analysis[J]. 2015, 49(1): 1-6.
DOI：

Spectral Element Method for Convection-Diffusion Equation with Stability Analysis[J]. 2015, 49(1): 1-6. DOI： 10.7652/xjtuxb201501001.

摘要

探讨了一维对流扩散方程的一种高精度数值解法

该解法在空间上采用了Chebyshev谱元方法

在时间上结合了半隐式Adams方法。通过数值算例验证了解法的可行性

利用特征分析法得到了对流扩散方程谱元求解时不同离散形式的稳定性条件

并对数值求解的稳定性进行了预测。通过时间步长、网格划分、对流项和黏性项插值阶数的影响研究表明:耦合Chebyshev谱元方法和半隐式Adams方法在求解对流扩散方程时能够获得高精度的数值解; 时间离散时Adams方法的黏性项采用一阶插值形式、对流项采用二阶插值形式

在未增加计算量的同时能够获得较大的稳定区域和较高的计算精度。

Abstract

The Chebyshev spectral element method combining with the semi-implicit Adams method is presented for solving the one-dimensional convection-diffusion equation

and the feasibility is verified by a numerical example. The stability condition of different discrete forms of spectral element method is deduced with character analysis

and the influences of time step

grid partitioning

interpolation order of convective and viscous terms are discussed. It is demonstrated that numerical solution with high accuracy can be gained with the coupled spectral element and semi-implicit Adams method for the convection-diffusion equation. Larger stability domain and higher accuracy can be achieved without extra-calculation as first-order viscous terms and second-order convective terms are interpolated in Adams method.

关键词

Keywords

references

GE L X, ZHANG J. High accuracy iterative solution of convection diffusion equation with boundary layers on nonuniform grids [J]. Journal of Computational Physics, 2001, 171(2): 560-578.

ZIENKIEWICZ O C, HEINRICH J C. The finite element method and convection problems in fluid mechanics [M]. New York, USA: John Wiley Sons Inc., 1978: 1-22.

GOTTLIEB D, ORSZAG S A. Numerical analysis of spectral method: theory and application [M]. Philadelphia, PA, USA: SIAM, 1977: 139-143.

MOFID A, PEYRET R. Stability of the Chebyshev collocation approximation to the advection-diffusion equation [J]. Computers Fluids, 1993, 22(4/5): 453-465.

PETERA A T. A spectral element method for fluid dynamics: laminar flow in a channel expansion [J]. Journal of Computational Physics, 1984, 54(3): 468-488.

FISCHER P, MULLEN J. Filter-based stabilization of spectral element methods [J]. Comptes Rendus de L'Académie des Sciences, 2001, 332(3): 265-270.

XU C J, PSAQUETTI R. Stabilized spectral element computations of high Reynolds number incompressible flows [J]. Journal of Computational Physics, 2004, 196(2): 680-704.

秦国良, 徐忠. 谱元方法求解二维不可压缩Navier-Stokes方程 [J]. 应用力学学报, 2000, 17(4): 20-26.

QIN Guoliang, XU Zhong. A spectral element method for incompressible Navier-Stokes equations [J]. Chinese Journal of Applied Mechanics, 2000, 17(4): 20-26. [9] 陈雪江, 秦国良, 徐忠. 谱元法和高阶时间分裂法求解方腔顶盖驱动流 [J]. 计算力学学报, 2002, 19(3): 281-285.

CHEN Xuejiang, QIN Guoliang, XU Zhong. Spectral element method and high order splitting method for Navies-Stokes equation [J]. Chinese Journal of Computational Mechanics, 2002, 19(3): 281-285.

WILLIAM GEAR C. Numerical initial value problem in ordinary differential equations [M]. Englewood Cliffs, NJ, USA: Prentice-Hall Inc., 1973: 147-158.

张家忠. 非线性动力系统的运动稳定性、分岔理论及其应用 [M]. 西安: 西安交通大学出版社, 2010: 12-19.

ERCILIA S. The controversial stability analysis [J]. Applied Mathematics and Computation, 2003, 145(2/3): 777-794.

VAN DORSSELAER J L M, HUNDSDORFER W. Stability estimates based on numerical ranges with an application to a spectral method [J]. BIT Numerical Mathematics, 1994, 34(2): 228-238.

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.2012,46(3):100-106.[doi:10.7652/xjtuxb201203018]

詹飞龙,张楚华.对流扩散方程的拟谱格式稳定性分析及扩稳方法.2012,46(3):113-118.[doi:10.7652/xjtux2012030 20]

孙旭,张家忠,黄科峰.基于弹簧近似的非结构化网格自适应处理方法.2010,44(9):104-108.[doi:10.7652/xjtuxb2010 09021]

章争荣,张湘伟.对流扩散方程的数值流形格式及其稳定性分析.2010,44(9):104-108.[doi:10.7652/xjtuxb201009021]

张荣欣,秦国良.用切比雪夫谱元方法求解均匀流场中的声传播问题.2009,43(7):120-124.[doi:10.7652/xjtuxb2009 07026]

朱昌允,秦国良,徐忠.谱元方法求解波动方程时显式与隐式差分方法的比较.2008,42(9):1142-1145.[doi:10.7652/xjtuxb200809017]

朱昌允,秦国良,徐忠.谱元方法求解波动方程及影响其数值精度的相关因素.2008,42(1):56-59.[doi:10.7652/xjtuxb 200801013]

许靖,秦国良,朱昌允.谱元方法求解含吸收边界的声学问题.2007,41(7):875-878.[doi:10.7652/xjtuxb200707027]

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

A Stabilized Spectral Element Method for Reaction-Convection-Diffusion Equation

A Hybrid-Topology Overlay Network for Supporting Large-Scale Application-Level Multicast

Dynamics in the Evolution of Spherical Micro-Bubble in Water and Its Control

Methods for Broadening the Band Gap Width of Local Resonance Elastic Metamaterial Beams

Effect of Control Delay on Dynamic Characteristics of Negative Stiffness Duffing System

Related Author

林静祥

艾子健

秦国良

2 董巍 1

刘晓 1

郑庆华 1

Related Institution

Ministry of Education Key Lab for Intelligent Networks and Network Security, Xi'an Jiaotong University

Department of Computer Science and Technology, Xi'an Jiaotong University

School of Energy and Power Engineering, Xi'an Jiaotong University

School of Mechanical Engineering, Xi'an Jiaotong University

Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi'an Jiaotong University

⁰