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一维非定常对流扩散方程的高阶组合紧致迎风格式

  

  1. 1. 复旦大学力学与工程科学系, 上海 200433;
    2. 宁夏大学数学计算机学院, 银川 750021
  • 收稿日期:2011-12-24 出版日期:2012-06-15 发布日期:2012-06-13
  • 基金资助:
    宁夏自然科学基金资助项目(NZ0938).

赵秉新. 一维非定常对流扩散方程的高阶组合紧致迎风格式[J]. 数值计算与计算机应用, 2012, 33(2): 138-148.

Zhao Bingxin. A HIGHT-ORDER COMBINED COMPACT UPWIND DIFFERENCE SCHEME FOR SOLVING 1D UNSTEADY CONVECTION-DIFFUSION EQUATION[J]. Journal of Numerical Methods and Computer Applications, 2012, 33(2): 138-148.

A HIGHT-ORDER COMBINED COMPACT UPWIND DIFFERENCE SCHEME FOR SOLVING 1D UNSTEADY CONVECTION-DIFFUSION EQUATION

Zhao Bingxin   

  1. 1. Department Mechanics and Engineering Science, Fudan University, Shanghai 200433, China;
    2. School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China
  • Received:2011-12-24 Online:2012-06-15 Published:2012-06-13
通过将对流项采用四五阶组合迎风紧致格式离散, 扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h4+τ4). 经Fourier精度分析和数值验证, 证实了格式的良好性能.三个数值算例包括线性常系数问题, 矩形波问题和非线性问题, 数值结果表明:该格式具有很高的分辨率, 且适用于对高雷诺数问题的数值模拟.
A fourth-order combined compact upwind (CCU) finite difference scheme was proposed for solving 1D unsteady convection-diffusion equation. Convection terms were discretized by combined fourth-order and fifth-order compact upwind schemes. Viscous terms were discretized by fourth-order compact symmetric finite difference scheme. After that, the semidiscretized equation was solved by fourth-order Runge-Kutta formula in time. The truncation error of the CCU scheme is O(h4 +τ4). Its excellent properties are proved by Fourier analyses and three numerical examples, which include linear and nonlinear convection-diffusion equations and rectangular wave problem. The results show that the CCU scheme is capable of capturing the minute physical changes for its high resolution, and is applicable to high Reynolds number problems.

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[1] Brandt A, Yavneh I. Inadequacy of first-order upwind difference schemes for some recirculating flows[J]. Journal of Computational Physics, 1991, 93(1): 128-143.
[2] Zhang J. Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number[J]. Numerical Methods for Partial Differential Equations, 1997, 13(1): 77-92.
[3] Tian Z F, Ge Y B. A fourth-order compact finite difference scheme for the steady stream functionvorticity formulation of the Navier-Stokes/Boussinesq equations[J]. International Journal for Numerical Methods in Fluids, 2003, 41(5): 495-518.
[4] Hirsh R S. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique[J]. Journal of Computational Physics, 1975, 19(1): 90-109.
[5] Ciment M, Leventhal S H, Weinberg B C. The operator compact implicit method for parabolic equations[J]. Journal of Computational Physics, 1978, 28(2): 135-166.
[6] Alain R. High order difference schemes for unsteady one-dimensional diffusion-convection problems[ J]. Journal of Computational Physics, 1994, 114(1): 59-76.
[7] Noye B J, Tan H H. A third-order semi-implicit finite difference method for solving the onedimensional convection-diffusion equation[J]. International Journal for Numerical Methods in Engineering, 1988, 26(7): 1615-1629.
[8] 陈国谦,高智. 对流扩散方程的迎风变换及相应有限差分方法[J].力学学报, 1991, 23(4): 418-425.
[9] Chen G Q, Gao Z, Yang Z F. A perturbational h4 exponential finite difference scheme for the convective diffusion equation[J]. Journal of Computational Physics, 1993, 104(1): 129-139.
[10] 王彩华. 一维对流扩散方程的一类新型高精度紧致差分格式[J]. 水动力学研究与进展(A辑), 2004, 19(5): 655-663.
[11] Ding H, Zhang Y. A new difference scheme with high accuracy and absolute stability for solving convection-diffusion equations[J]. Journal of Computational and Applied Mathematics, 2009, 230(2):600-606.
[12] Tian Z F, Yu P X. A high-order exponential scheme for solving 1D unsteady convection-diffusion equations[J]. Journal of Computational and Applied Mathematics, 2011, 235(8): 2477-2491.
[13] Lele S K. Compact finite difference schemes with Spectral-like resolution[J]. Journal of Computational Physics, 1992, 103(1): 16-42.
[14] 田振夫. 高精度紧致差分方法及其应用研究[D]. 上海: 上海大学, 2006.
[15] 梁贤,田振夫. 求解Navier-Stokes方程组的组合紧致迎风格式[J].计算物理, 2008, 25(6): 659-667.
[16] Ma Y W, Fu D X, Kobayashi T,et al. Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme[J]. International Journal for Numerical Methods in Fluids, 1999, 30(5): 509-521.
[17] Tian Z F, Li Y A. Numerical solution of the incompressible Navier-Stokes equations with a threepoint fourth-order upwind compact difference scheme[C]. ICNM-IV,Shanghai, 2002: 942-946.
[18] Kim J W, Lee D J. Optimized compact finite difference schemes with maximum resolution[J]. AIAA Journal, 1996, 34: 887-893.
[19] Gustafsson B. The convergence rate for difference approximations to mixed initial boundary value problems[J]. Mathematics of Computation, 1975, 29(130): 396-406.
[20] 高智. 对流扩散方程的绝对稳定高阶中心差分格式[J]. 力学学报, 2010, 42(5): 811-817.
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