对流扩散方程的差分解法
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摘要
根据已发展的二阶微商三次样条四阶逼近公式,提出了基于线性插值的求解对流扩散方程的特征差分格式。通过 Fourier 方法讨论了该格式的稳定性。数值结果表明,本文对对流扩散方程给出的特征差分格式明显优于一般的基于线性插值的特征差分格式。
利用第二类 Saul’yev 非对称格式给出了对流扩散方程的一类交替分组显格式。该方法具有并行本性,并且绝对稳定。数值结果表明,本文对对流扩散方程给出的 AGE 算法明显优于 Evans 和 Abdullah[15]所提出的交替分组显格式,因此本文方法是一种有效算法。
将特征线法和有限差分法相结合,借助于斜线性插值,分别给出了求解线性和非线性对流占优扩散方程的一种新的特征差分格式,并研究了算法的收敛性。该算法的优点是特别适用于求解变系数的对流占优扩散方程,能更有效地消除数值振荡现象。
A new kind of characteristic-difference scheme is proposed for solving convecti -on diffusion equations with the linear interpolation method.The method is based on the cubic-spline difference formular of fourth-order accuracy for second order deriva -tives developed byTian Zhenfu.The stability of the characteristic difference scheme is studied.The results prove that our method is better than the ordinary characteristic difference scheme with the linear interpolation method.
Based on the Saul’yev asymmetric schemes, a new alternating group explicit (AGE)method for solving convection-diffusion equation is derived in this paper. The method has the obvious property of parallelism,and is unconditionally stable. Numer -ical example is given which illustrated that the present method is in preference to Evans and Abdullah’AGE method for solving the convection-diffusion equation in [15].
A new kind of characteristic-difference scheme for convection-dominated diffus -ion equations is constructed by combining characteristic method with the finite -difference method and with the skew linear interpolation method. The convergence of the characteristic-difference scheme is studied. The advantage of this scheme is very adaptable to obtain the solution of the equations with variable coefficient and can eliminate the numerical oscillations more efficiently.
利用第二类 Saul’yev 非对称格式给出了对流扩散方程的一类交替分组显格式。该方法具有并行本性,并且绝对稳定。数值结果表明,本文对对流扩散方程给出的 AGE 算法明显优于 Evans 和 Abdullah[15]所提出的交替分组显格式,因此本文方法是一种有效算法。
将特征线法和有限差分法相结合,借助于斜线性插值,分别给出了求解线性和非线性对流占优扩散方程的一种新的特征差分格式,并研究了算法的收敛性。该算法的优点是特别适用于求解变系数的对流占优扩散方程,能更有效地消除数值振荡现象。
A new kind of characteristic-difference scheme is proposed for solving convecti -on diffusion equations with the linear interpolation method.The method is based on the cubic-spline difference formular of fourth-order accuracy for second order deriva -tives developed byTian Zhenfu.The stability of the characteristic difference scheme is studied.The results prove that our method is better than the ordinary characteristic difference scheme with the linear interpolation method.
Based on the Saul’yev asymmetric schemes, a new alternating group explicit (AGE)method for solving convection-diffusion equation is derived in this paper. The method has the obvious property of parallelism,and is unconditionally stable. Numer -ical example is given which illustrated that the present method is in preference to Evans and Abdullah’AGE method for solving the convection-diffusion equation in [15].
A new kind of characteristic-difference scheme for convection-dominated diffus -ion equations is constructed by combining characteristic method with the finite -difference method and with the skew linear interpolation method. The convergence of the characteristic-difference scheme is studied. The advantage of this scheme is very adaptable to obtain the solution of the equations with variable coefficient and can eliminate the numerical oscillations more efficiently.
引文
[1] P.C. Chatwin, C.M. Allen, Mathematical models of dispersion in rivers and estuaries, Ann. Rev. Fluid Mech. 17 (1985) 119–149.
[2] M.H. Chaudhry, D.E. Cass, J.E. Edinger, Modelling of unsteady-flow water temperatures, J. Hydraul. Eng. 109 (5) (1983) 657–669.
[3] Q.N. Fattah, J.A. Hoopes, Dispersion in anisotropic homogeneous porous media, J. Hydraul. Eng. 111 (1985) 810–827.
[4] C.R. Gane, P.L. Stephenson, An explicit numerical method for solving transient combined heat conduction and convection problems, Int. J. Numer. Meth. Eng. 14 (1979) 1141–1163.
[5] V. Guvanasen, R.E. Volker, Numerical solutions for solute transport in unconfined aquifers, Int. J. Numer. Meth. Fluids 3 (1983) 103–123.
[6] B.J. Noye, Numerical solution of partial differential equations, Lecture Notes, 1990.
[7] 杨瑞琰.对流扩散方程有限混合分析格式[J].大学数学,2003,19(6):102-104.
[8] 许延生.数值求解非线性 Burgers 方程的有限分析混合格式[J].水动力学研究与进展 1998,13(2):162-166.
[9] 陈和春,李炜.基于混合有限分析插值的非交错网格方法[J].三峡大学学报,2003,25(4):301-304.
[10] 赵明登,郑邦民. 三维对流扩散方程的混合有限分析解[J].水利学报,1995(9):55-62.
[11] 李炜,彭文启. 线性对流扩散方程混合有限分析多重网格法[J].武汉电力大学学报,1995,28(2):215-219.
[12] 杨瑞琰. 对流扩散方程有限混合分析格式的稳定性分析[J].陕西工学院学报,2001,17(4):54-58.
[13] 张文博,孙澈.线性定常对流占优对流扩散问题的有限体积——流线扩散有限元法[J].计算数学, 2004,(1): 93-108.
[14] 高智, 柏威.对流扩散方程的摄动有限体积(PFV)方法及讨论[J].力学学报,2004,(1): 88-93.
[15] 窦红.对流扩散方程的一种显式有限体积-有限元方法[J].应用数学与计算数学学报, 200 1,(2):45-52.
[16] Douglas J,Russell T F.Numerical methods for convection-dominated diffusion problem based on combining the method of characteristics with finite element or finite difference procedures [J]. SIAM J Numer Anal,1982,19(5):871-885.
[17] R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys.1974,14(2):159-179.
[18] Mehdi Dehghan, Weighted finite differentce techniques for the one-dimensional advection diffusion equation, Applied Mathematics and Computation. 2004,147(2):307-319.
[19] Mehdi Dehghan, Numerical solution of the three-dimensional advection-diffusion equation, Applied Mathematics and Computation.2004,150(1):5-19.
[20] A. Kolesnikov, A.J. Baker.Efficient implementation of high order methods for the Advection diffusion equation ,Comput. Methods Appl. Mech. Engrg.2000,189:701-722.
[21] 陆金甫,杜正平,刘晓遇.对流占优扩散问题的一种特征差分方法[J].清华大学学报(自然科学版),2002,42(8):35-37.
[22] 田振夫.泊松方程的高精度三次样条差分方法.西北师范大学学报(自然科学版),1996,32(2):13-17.
[23] D.J.Evans,A.R.Abdullah.A New Explicit Method for the Diffusion-convection Equations[J]. Comp and Math with Appl,1985,11:145-154.
[24] D.J.Evans,A.R.Abdullah.Alternating Group Explicit Methods for the Diffusion Equations[J]. Appli Math Modelling,1985,9:201-206.
[25] 陆金甫,张宝琳,徐涛.求解对流扩散方程的交替分段显-隐式方法[J].数值计算与计算机应用,1998,19:161-167.
[26] 张宝琳,符鸿源.一类交替块 Crank-Nicolson 方法的差分图[J].科学通报,1999,40 (11):1148-1152.
[27] 王文洽.变系数对流扩散方程的交替分段显-隐式方法[J].山东大学学报,2002,37(2): 120-123.
[28] 王文洽.变系数对流-扩散方程的交替分段 Crank-Nicolson 方法[J].应用数学和力学,2003,24(1):29-38.
[29] 曾文平.对流-扩散方程的两类交替方法之比较[J].华侨大学学报(自然科学版),1999,20(2):118-122.
[30] 郑兴华.二维对流扩散方程的分步交替分组显式格式[J].华侨大学学报(自然科学版),200 2,23(2):122-128.
[31] 刘晓遇.解对流扩散方程的显式交替方向法[J].清华大学学报(自然科学版),1999,39(12): 31-35.
[32] 王文洽.求解扩散方程的一类交替分组显式方法[J].山东大学报,2002,37(3):194-199.
[33] Saul’yev V K.Integration of Techniques for Fluid Dynamics[M] .Berlin:Springe-Verlag,1998.
[34] Kellog R B.An alternating direction method for operator equations [J].SIAM, 1964,12(6): 848-854.
[35] 杜正平,刘晓遇,陆金甫.对流扩散方程的二次单调插值特征差分方法[J].清华大学学报(自然科学版),2000,40(11):1-4.
[36] 秦新强,马逸尘.对流占优扩散方程的改进特征差分算法[J]. 工程数学学报,2002,19(4):51-56.
[37] 秦新强,李秋芳.对流占优扩散方程一种新的特征差分算法[J].高等学校计算数学学报,2004,19:161-167.
[38] 由同顺,孙澈.求解非线性对流-扩散问题的特征差分方法[J].计算数学,1991,13(2):166-176.
[39] 由同顺,孙澈.非线性对流-扩散方程初边值问题的特征-差分解法[J].计算数学,1993,15(2):143-155.
[40] 由同顺.二维非线性对流-扩散方程的特征-差分解法[J].计算数学,1993,15(4):402-409.
[41] 由同顺.非线性对流-扩散方程高阶特征差分格式及其误差估计[J].数值计算与计算机应用,1994,15(4):312-317.
[42] 由同顺.二维非线性对流-扩散方程的非震荡特征差分方法[J].计算数学,2000,22(2):159- 166.
[43] 由同顺.求解对流扩散方程的一支高精度非震荡特征差分方法[J].工程数学学报,2002,19(2):57-62.
[44] 由同顺. 求解对流扩散方程的高阶 ICT-MMOCAA 差分方法[J].应用数学,2002,15(2): 132-136.
[45] 谢树森,丁婉.解对流扩散方程的修正特征差分方法[J].青岛化工学院学报,2002,23(2):9- 13.
[46] 穆祖元.解对流扩散方程的沿特征线中心差分格式[J].同济大学学报,1997,25(3):354- 359.
[47] Fletcher CAJ,Computational techniques for fluid dynamics 1,Berlin:Springer-Verlag, 1988,331-374.
[2] M.H. Chaudhry, D.E. Cass, J.E. Edinger, Modelling of unsteady-flow water temperatures, J. Hydraul. Eng. 109 (5) (1983) 657–669.
[3] Q.N. Fattah, J.A. Hoopes, Dispersion in anisotropic homogeneous porous media, J. Hydraul. Eng. 111 (1985) 810–827.
[4] C.R. Gane, P.L. Stephenson, An explicit numerical method for solving transient combined heat conduction and convection problems, Int. J. Numer. Meth. Eng. 14 (1979) 1141–1163.
[5] V. Guvanasen, R.E. Volker, Numerical solutions for solute transport in unconfined aquifers, Int. J. Numer. Meth. Fluids 3 (1983) 103–123.
[6] B.J. Noye, Numerical solution of partial differential equations, Lecture Notes, 1990.
[7] 杨瑞琰.对流扩散方程有限混合分析格式[J].大学数学,2003,19(6):102-104.
[8] 许延生.数值求解非线性 Burgers 方程的有限分析混合格式[J].水动力学研究与进展 1998,13(2):162-166.
[9] 陈和春,李炜.基于混合有限分析插值的非交错网格方法[J].三峡大学学报,2003,25(4):301-304.
[10] 赵明登,郑邦民. 三维对流扩散方程的混合有限分析解[J].水利学报,1995(9):55-62.
[11] 李炜,彭文启. 线性对流扩散方程混合有限分析多重网格法[J].武汉电力大学学报,1995,28(2):215-219.
[12] 杨瑞琰. 对流扩散方程有限混合分析格式的稳定性分析[J].陕西工学院学报,2001,17(4):54-58.
[13] 张文博,孙澈.线性定常对流占优对流扩散问题的有限体积——流线扩散有限元法[J].计算数学, 2004,(1): 93-108.
[14] 高智, 柏威.对流扩散方程的摄动有限体积(PFV)方法及讨论[J].力学学报,2004,(1): 88-93.
[15] 窦红.对流扩散方程的一种显式有限体积-有限元方法[J].应用数学与计算数学学报, 200 1,(2):45-52.
[16] Douglas J,Russell T F.Numerical methods for convection-dominated diffusion problem based on combining the method of characteristics with finite element or finite difference procedures [J]. SIAM J Numer Anal,1982,19(5):871-885.
[17] R.F. Warming, B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys.1974,14(2):159-179.
[18] Mehdi Dehghan, Weighted finite differentce techniques for the one-dimensional advection diffusion equation, Applied Mathematics and Computation. 2004,147(2):307-319.
[19] Mehdi Dehghan, Numerical solution of the three-dimensional advection-diffusion equation, Applied Mathematics and Computation.2004,150(1):5-19.
[20] A. Kolesnikov, A.J. Baker.Efficient implementation of high order methods for the Advection diffusion equation ,Comput. Methods Appl. Mech. Engrg.2000,189:701-722.
[21] 陆金甫,杜正平,刘晓遇.对流占优扩散问题的一种特征差分方法[J].清华大学学报(自然科学版),2002,42(8):35-37.
[22] 田振夫.泊松方程的高精度三次样条差分方法.西北师范大学学报(自然科学版),1996,32(2):13-17.
[23] D.J.Evans,A.R.Abdullah.A New Explicit Method for the Diffusion-convection Equations[J]. Comp and Math with Appl,1985,11:145-154.
[24] D.J.Evans,A.R.Abdullah.Alternating Group Explicit Methods for the Diffusion Equations[J]. Appli Math Modelling,1985,9:201-206.
[25] 陆金甫,张宝琳,徐涛.求解对流扩散方程的交替分段显-隐式方法[J].数值计算与计算机应用,1998,19:161-167.
[26] 张宝琳,符鸿源.一类交替块 Crank-Nicolson 方法的差分图[J].科学通报,1999,40 (11):1148-1152.
[27] 王文洽.变系数对流扩散方程的交替分段显-隐式方法[J].山东大学学报,2002,37(2): 120-123.
[28] 王文洽.变系数对流-扩散方程的交替分段 Crank-Nicolson 方法[J].应用数学和力学,2003,24(1):29-38.
[29] 曾文平.对流-扩散方程的两类交替方法之比较[J].华侨大学学报(自然科学版),1999,20(2):118-122.
[30] 郑兴华.二维对流扩散方程的分步交替分组显式格式[J].华侨大学学报(自然科学版),200 2,23(2):122-128.
[31] 刘晓遇.解对流扩散方程的显式交替方向法[J].清华大学学报(自然科学版),1999,39(12): 31-35.
[32] 王文洽.求解扩散方程的一类交替分组显式方法[J].山东大学报,2002,37(3):194-199.
[33] Saul’yev V K.Integration of Techniques for Fluid Dynamics[M] .Berlin:Springe-Verlag,1998.
[34] Kellog R B.An alternating direction method for operator equations [J].SIAM, 1964,12(6): 848-854.
[35] 杜正平,刘晓遇,陆金甫.对流扩散方程的二次单调插值特征差分方法[J].清华大学学报(自然科学版),2000,40(11):1-4.
[36] 秦新强,马逸尘.对流占优扩散方程的改进特征差分算法[J]. 工程数学学报,2002,19(4):51-56.
[37] 秦新强,李秋芳.对流占优扩散方程一种新的特征差分算法[J].高等学校计算数学学报,2004,19:161-167.
[38] 由同顺,孙澈.求解非线性对流-扩散问题的特征差分方法[J].计算数学,1991,13(2):166-176.
[39] 由同顺,孙澈.非线性对流-扩散方程初边值问题的特征-差分解法[J].计算数学,1993,15(2):143-155.
[40] 由同顺.二维非线性对流-扩散方程的特征-差分解法[J].计算数学,1993,15(4):402-409.
[41] 由同顺.非线性对流-扩散方程高阶特征差分格式及其误差估计[J].数值计算与计算机应用,1994,15(4):312-317.
[42] 由同顺.二维非线性对流-扩散方程的非震荡特征差分方法[J].计算数学,2000,22(2):159- 166.
[43] 由同顺.求解对流扩散方程的一支高精度非震荡特征差分方法[J].工程数学学报,2002,19(2):57-62.
[44] 由同顺. 求解对流扩散方程的高阶 ICT-MMOCAA 差分方法[J].应用数学,2002,15(2): 132-136.
[45] 谢树森,丁婉.解对流扩散方程的修正特征差分方法[J].青岛化工学院学报,2002,23(2):9- 13.
[46] 穆祖元.解对流扩散方程的沿特征线中心差分格式[J].同济大学学报,1997,25(3):354- 359.
[47] Fletcher CAJ,Computational techniques for fluid dynamics 1,Berlin:Springer-Verlag, 1988,331-374.