扩散方程和对流扩散方程的高效率差分格式及其并行算法
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摘要
本论文就扩散方程和对流扩散方程，在乘积型差商空间中利用待定系数法构造了一系列高精度、高稳定性的差分格式，并给出相应的数值例子，在第四章构造了求解对流扩散方程的一组高效率并行GE、GER、GEL及AGE格式，相应的数值例子验证了理论分析的正确性。
本论文主要分为以下四个部分：
第一部分：首先介绍目前国内外科研工作者在研究数值求解偏微分方程中所运用的一些基本研究思想和方法及已取得的一些成果，然后介绍有关偏微分方程有限差分方法的基本概念和原理。
第二部分：运用待定系数法构造了求解一维扩散方程的两层七点半显差分格式，格式的截断误差达到O(τ~2+h~4)，稳定性条件是0＜r≤0.4647。其精度优于已有的求解扩散方程的半显格式，且在计算中是显式计算，速度快，效率高。
第三部分：再次利用待定系数法给出了求解对流扩散方程的一个两层五点半显格式，对初边值问题可显式计算，计算简单。格式的截断误差达到O(τ~2+h~2)，并具有较好的稳定性。
第四部分：针对对流扩散方程，利用文[12](张大凯等，1991年)中给出的一组saul'yev型非对称格式构造了一类并行GE格式，在空间节点为奇数时，构造了GEL和GER格式。上述格式的截断误差为O(τ+h)，稳定性条件为r≤h/ε＜2/|α|，再利用GER和GEL格式，构造了绝对稳定的交替分组显式AGE格式，此格式并行度好，计算速度快，效率高。
In this paper we report a class of high-precise、high-stability difference schemes in the product difference space and it applies to the one-order one-dimensional constant coefficient parabolic equation.The last we give a high-efficiency parallel algorithm. In this chapter every type scheme is given numerical compute,and the results validate the theory analysis.
There are four parts in this paper:
The first, we give some basic definitions and some lemmas about combination difference algorithm,we analyze the relation of node distributing and type of difference schemes, then bring forward the undetermined coefficient principle in the product difference space.
The second, a semi-explicit differencing scheme of seven points for solving the parabolic equations of one-dimensions is presented,The truncation error isO(τ~2 + h~4) ,and the stability condition is 0 < r≤0.4647.
The third, a semi-explicit different scheme of five points for solving the diffusion convection equation is presented in this paper. It can be calculated explicitly forinitial-boundary value problems, The truncation error is O(τ~2 +h~2) .and thestability is well.
The last, we give a high-efficiency parallel algorithm, the schemetruncation error is O(τ+ h) and the stability condition is r≤h/ε< 2/|α| .the schemeis given numerical compute,and the results validate the theory analysis.
本论文主要分为以下四个部分：
第一部分：首先介绍目前国内外科研工作者在研究数值求解偏微分方程中所运用的一些基本研究思想和方法及已取得的一些成果，然后介绍有关偏微分方程有限差分方法的基本概念和原理。
第二部分：运用待定系数法构造了求解一维扩散方程的两层七点半显差分格式，格式的截断误差达到O(τ~2+h~4)，稳定性条件是0＜r≤0.4647。其精度优于已有的求解扩散方程的半显格式，且在计算中是显式计算，速度快，效率高。
第三部分：再次利用待定系数法给出了求解对流扩散方程的一个两层五点半显格式，对初边值问题可显式计算，计算简单。格式的截断误差达到O(τ~2+h~2)，并具有较好的稳定性。
第四部分：针对对流扩散方程，利用文[12](张大凯等，1991年)中给出的一组saul'yev型非对称格式构造了一类并行GE格式，在空间节点为奇数时，构造了GEL和GER格式。上述格式的截断误差为O(τ+h)，稳定性条件为r≤h/ε＜2/|α|，再利用GER和GEL格式，构造了绝对稳定的交替分组显式AGE格式，此格式并行度好，计算速度快，效率高。
In this paper we report a class of high-precise、high-stability difference schemes in the product difference space and it applies to the one-order one-dimensional constant coefficient parabolic equation.The last we give a high-efficiency parallel algorithm. In this chapter every type scheme is given numerical compute,and the results validate the theory analysis.
There are four parts in this paper:
The first, we give some basic definitions and some lemmas about combination difference algorithm,we analyze the relation of node distributing and type of difference schemes, then bring forward the undetermined coefficient principle in the product difference space.
The second, a semi-explicit differencing scheme of seven points for solving the parabolic equations of one-dimensions is presented,The truncation error isO(τ~2 + h~4) ,and the stability condition is 0 < r≤0.4647.
The third, a semi-explicit different scheme of five points for solving the diffusion convection equation is presented in this paper. It can be calculated explicitly forinitial-boundary value problems, The truncation error is O(τ~2 +h~2) .and thestability is well.
The last, we give a high-efficiency parallel algorithm, the schemetruncation error is O(τ+ h) and the stability condition is r≤h/ε< 2/|α| .the schemeis given numerical compute,and the results validate the theory analysis.
引文
[1] 马明书，解抛物型方程的一个高精度两层显格式[J]，河南师范大学学报，1996．1
[2] 马明书，解抛物型方程的一族高精度差分格式[J]，高等学校计算数学学报，1996．2
[3] 马明书，王同科，解抛物型方程的一个高精度隐式差分格式[J]，纺织高校基础科学学报，2000．3
[4] 田振夫，冯秀芳，对流扩散方程的一种新的显式方法[J]，工程数学学报，2000．4
[5] 田振夫，含源定常对流扩散方程的一种高精度差分格式[J]，宁夏大学学报，1995．1
[6] 李荣华，冯果忱。微分方程数值解法[M]，高等教育出版社，1995，5
[7] 李庆扬等，现代数值分析[M]，高等教育出版社，1995，10
[8] 李物兰，张大凯，求解对流扩散方程的一种新的隐格式[J]，贵州大学学报，2004．4
[9] 陈泽龙，张大凯，求解抛物型方程的七点半显格式[J]，贵州大学学报，2006．2
[10] 陆金甫，关治，偏微分方程数值解[M]，清华大学出版社，2004，1
[11] 张宝琳，谷同祥，莫则尧，数值并行计算原理与方法[M]，国防工业出版社，1999，
[12] 张大凯，李洪林，求解对流扩散方程的两层半显差分格式[J]，贵州科学，1991．1
[13] 张天德，关于解(?)u／(?)t=σ(?)~2u／(?)x~2的差分格式[J]，山东工业大学学报，2000．2
[14] 张莉，张大凯，解抛物型方程的组合差商法[J]，贵州大学学报，2004．4
[15] 杨志峰，陈国谦，对流扩散方程经典中心差分格式的改进[J]，北京师范大学学报，1994．1
[16] 武蔚文，张大凯，对流扩散方程的差分格式[J]，计算机应用，1989．2
[17] 金承日，解抛物型方程的高精度显式差分格式[J]，计算数学，1991．1
[18] 明祖芬，求解对流扩散方程的两个半显差分格式[J]，贵州大学学报，1991．1
[19] 胡家赣，线性代数方程组的迭代解法[M]，科学出版社，1991
[20] 胡健伟，汤怀民，微分方程数值解法[M]，科学出版社，1999，1
[21] 郭本瑜，偏微分方程的差分方法[M]，科学出版社，1988，2
[22] 曹俊英，张大凯，解抛物型方程的九点隐格式[J]，贵州大学学报，2006．5
[23] 梁洁，求解隐式差分方程的高精度并行算法[J]，贵州大学学报，2004．1
[24] 曾晓艳，陈建业，孙乐林，对流扩散方程的一种新型差分格式[J]，数学杂志，2003．1
[25] 戴嘉尊，邱建贤，微分方程数值解法[M]，东南大学出版社，2002，2
[26] Amodio P and Mastronardi N,A parallel version of the cyclic reduction algorithm on a hypercube. Parallel Computing[M], 1993, 19
[27] David Gilbarg Neils Trudinger Elliptic Partial Differential Equations of second Order[M]. 1983
[28] Evans D J, Abdullah A R B. A new explicit method for the diffusion-convection equation. Comp. & Maths. with Appls[J], 1985, 11
[29] Evans D. J Alternating group explicit method for hyperbolic equations Compute[J]. Math. Appl. 1988. 15
[30] Gu Tongxiang Asynchronous relaxed iterative methods for solving linear systems of equations[J], Appl. Math. Meth, 1997. 18 (8)
[31] J. M. Varah A lower bound for the smallest slngular value of a matrix[J], Lin. Alg. Appl, 11, 3-5(1975).
[32] Liu Qingfu, Zhong Weijun, Parallel EQI algorithm with different insertionschemes[J], JournalofSoutheastUniversity, sep2003, VOL. 19
[33] Lu Jinfu, Zhang Baolin, Modified AGE method for the convection diffusion equation. Commun. Numer. Methods in Engin, 1998. 14
[34] Mehrmann V. Divide and conqer methods for tridiagonal linear systems. in: Notes on Numerical Fluid Mechanics[J]. Proc of the Sixth GAMM-Seminar, Kiel (jan. 19-21 1990) Wolfgang Hackbusch (Editor), 1991.
[35] Michielse P H and Van der Vorst H A, Data transport in Wang's partition methom[J]. Parallel processing, 1988, 7
[36] Richard J. Babarsky and Robert Sharpley. Expanded stability through higher accuracy for time-centered advection schemes[J]. June 1997 American Meteorological Society.
[37] R. K. Mohanty and Urvashi Arora, A new discretization method of order four for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equation[J], International Journal of Mathematical education in Science and Technology, 2002
[38] R. K. Mohanty. An operator splitting technique for anunconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable codfficients[J]. Applied Mathematics and Computation. 2005.162
[39] R.K. Moiianty. An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation[J], Applied Mathematics Letters, 2004, 17
[40] R.S. Varga, On diagonal dominance argument for bounding[J], Lin, Alg. Appl. 14, 1976.
[41] Spalett G and Evans D J, The parallel recursive decouping algorithm for solving tridiagonal linear systems[J]. Parallel Computing, 1993, 19
[42] Sun X H. On the parallel diagonal dominant algorithm[J]. 1993 International Conference on Parallel processing. 1993, 10-17.
[43] Sun X H. Zhang H, and Ni L. Efficient tridiagonal solvers on multlcomputers[J]. IEEE Trans on Computers, 1992, 41(3): 286-296.
[44] Wang H H. A parallel method for triagonal equations[J]. ACM Trans. Math. Software, 1981, 7: 170-183.
[45] You Yongshun, three—step UNO—MMOCAA Difference Method for convection Diffusion equation[J], Northeast. MATH. J. 2004.1
[2] 马明书，解抛物型方程的一族高精度差分格式[J]，高等学校计算数学学报，1996．2
[3] 马明书，王同科，解抛物型方程的一个高精度隐式差分格式[J]，纺织高校基础科学学报，2000．3
[4] 田振夫，冯秀芳，对流扩散方程的一种新的显式方法[J]，工程数学学报，2000．4
[5] 田振夫，含源定常对流扩散方程的一种高精度差分格式[J]，宁夏大学学报，1995．1
[6] 李荣华，冯果忱。微分方程数值解法[M]，高等教育出版社，1995，5
[7] 李庆扬等，现代数值分析[M]，高等教育出版社，1995，10
[8] 李物兰，张大凯，求解对流扩散方程的一种新的隐格式[J]，贵州大学学报，2004．4
[9] 陈泽龙，张大凯，求解抛物型方程的七点半显格式[J]，贵州大学学报，2006．2
[10] 陆金甫，关治，偏微分方程数值解[M]，清华大学出版社，2004，1
[11] 张宝琳，谷同祥，莫则尧，数值并行计算原理与方法[M]，国防工业出版社，1999，
[12] 张大凯，李洪林，求解对流扩散方程的两层半显差分格式[J]，贵州科学，1991．1
[13] 张天德，关于解(?)u／(?)t=σ(?)~2u／(?)x~2的差分格式[J]，山东工业大学学报，2000．2
[14] 张莉，张大凯，解抛物型方程的组合差商法[J]，贵州大学学报，2004．4
[15] 杨志峰，陈国谦，对流扩散方程经典中心差分格式的改进[J]，北京师范大学学报，1994．1
[16] 武蔚文，张大凯，对流扩散方程的差分格式[J]，计算机应用，1989．2
[17] 金承日，解抛物型方程的高精度显式差分格式[J]，计算数学，1991．1
[18] 明祖芬，求解对流扩散方程的两个半显差分格式[J]，贵州大学学报，1991．1
[19] 胡家赣，线性代数方程组的迭代解法[M]，科学出版社，1991
[20] 胡健伟，汤怀民，微分方程数值解法[M]，科学出版社，1999，1
[21] 郭本瑜，偏微分方程的差分方法[M]，科学出版社，1988，2
[22] 曹俊英，张大凯，解抛物型方程的九点隐格式[J]，贵州大学学报，2006．5
[23] 梁洁，求解隐式差分方程的高精度并行算法[J]，贵州大学学报，2004．1
[24] 曾晓艳，陈建业，孙乐林，对流扩散方程的一种新型差分格式[J]，数学杂志，2003．1
[25] 戴嘉尊，邱建贤，微分方程数值解法[M]，东南大学出版社，2002，2
[26] Amodio P and Mastronardi N,A parallel version of the cyclic reduction algorithm on a hypercube. Parallel Computing[M], 1993, 19
[27] David Gilbarg Neils Trudinger Elliptic Partial Differential Equations of second Order[M]. 1983
[28] Evans D J, Abdullah A R B. A new explicit method for the diffusion-convection equation. Comp. & Maths. with Appls[J], 1985, 11
[29] Evans D. J Alternating group explicit method for hyperbolic equations Compute[J]. Math. Appl. 1988. 15
[30] Gu Tongxiang Asynchronous relaxed iterative methods for solving linear systems of equations[J], Appl. Math. Meth, 1997. 18 (8)
[31] J. M. Varah A lower bound for the smallest slngular value of a matrix[J], Lin. Alg. Appl, 11, 3-5(1975).
[32] Liu Qingfu, Zhong Weijun, Parallel EQI algorithm with different insertionschemes[J], JournalofSoutheastUniversity, sep2003, VOL. 19
[33] Lu Jinfu, Zhang Baolin, Modified AGE method for the convection diffusion equation. Commun. Numer. Methods in Engin, 1998. 14
[34] Mehrmann V. Divide and conqer methods for tridiagonal linear systems. in: Notes on Numerical Fluid Mechanics[J]. Proc of the Sixth GAMM-Seminar, Kiel (jan. 19-21 1990) Wolfgang Hackbusch (Editor), 1991.
[35] Michielse P H and Van der Vorst H A, Data transport in Wang's partition methom[J]. Parallel processing, 1988, 7
[36] Richard J. Babarsky and Robert Sharpley. Expanded stability through higher accuracy for time-centered advection schemes[J]. June 1997 American Meteorological Society.
[37] R. K. Mohanty and Urvashi Arora, A new discretization method of order four for the numerical solution of one-space dimensional second-order quasi-linear hyperbolic equation[J], International Journal of Mathematical education in Science and Technology, 2002
[38] R. K. Mohanty. An operator splitting technique for anunconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable codfficients[J]. Applied Mathematics and Computation. 2005.162
[39] R.K. Moiianty. An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation[J], Applied Mathematics Letters, 2004, 17
[40] R.S. Varga, On diagonal dominance argument for bounding[J], Lin, Alg. Appl. 14, 1976.
[41] Spalett G and Evans D J, The parallel recursive decouping algorithm for solving tridiagonal linear systems[J]. Parallel Computing, 1993, 19
[42] Sun X H. On the parallel diagonal dominant algorithm[J]. 1993 International Conference on Parallel processing. 1993, 10-17.
[43] Sun X H. Zhang H, and Ni L. Efficient tridiagonal solvers on multlcomputers[J]. IEEE Trans on Computers, 1992, 41(3): 286-296.
[44] Wang H H. A parallel method for triagonal equations[J]. ACM Trans. Math. Software, 1981, 7: 170-183.
[45] You Yongshun, three—step UNO—MMOCAA Difference Method for convection Diffusion equation[J], Northeast. MATH. J. 2004.1