一类非线性偏微分方程组的差分方法及在污水处理模拟中的应用
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摘要
物理、化学、生物、地质等领域的很多模型都可归结为线性或非线性偏微分方程的定解问题。对于一个非线性偏微分方程，要想求出其精确解析解是相当困难的，于是只好转而去求其数值解。虽然偏微分方程数值解法的发展由来已久，但发展一直很慢，主要是由于人们受计算能力的限制。但直到20世纪70年代以来，随着计算机的高速发展，人们的计算能力大大增强，研究热点也逐渐转向非线性科学领域，非线性偏微分方程的数值解问题便是其中很重要的一方面。
人们在实践当中，对于微分方程定解问题，发展了很多方法，包括方法、各种混合元、边界元、无限元法及上世纪八十年代兴起的小波分析法，但应用最多的还是有限元法和有限差分法。
其中有限差分法的思想就是直接用各阶差商来代替微分方程中的各阶导数，根据具体问题和所选择的差分格式的不同，加上适当的初边值条件，便形成一个线性或非线性的代数方程组。差分方法的首要问题就是构造合理的差分格式，使得它的解保持原问题的某些主要性质，并且又足够精确。一个好的和实用的差分格式应满足相容性、稳定性和收敛性。
本文主要考虑一类如下形式的具有代表性的非线性耦合偏微分方程组初边值问题：
，
，
其中：
是维未知向量函数,是阶正定系数矩阵，是维向量函数，其中和属于矩形域：
是维已知
向量函数。
针对上述形式微分方程定解问题，构造如下的差分格式
，，
在适当的条件下给出稳定性的证明。
对于上述定解问题，若采用隐式差分格式，便得到一个非线性的差分方程组，这就涉及到非线性方程组数值解法问题，于是在第三章中简要介绍了非线性方程组的迭代方法，并针对如下形式的阻尼参数型迭代格式，
，
给出并证明了其收敛性的的充要条件。
最后，利用前述格式，实际应用当中我们对污水处理过程中的三氮进行了数值模拟。结果表明，数值解具有较好的收敛性和模拟效果，从而说明了所采用的差分格式和求解非线性差分方程的阻尼参数下降差分格式具有较高的实用价值。
Many mathematical models in the fields of physics, chemistry, biology and geology can be boiled down to the fixed solution problem of linear or nonlinear partial differential equations (PDE.). It’s so hard for people to get the accurate solution of nonlinear PDE as people have to switch to the numerical solution. Despite the long history of the numerical solution method of PDE, it had advance little due to the restraints of people’s computing ability. It is not until the 70s of the 20th, with the rapid development of computer, people has much strengthened his computing ability and focused the research on the nonlinear science which includes the important aspect of numerical solution of PDE.
In practice, people has developed many kinds of numerical solution methods which covers the Galerkin method and all kinds of mixed or boundary or infinite element methods as well as the wavelet method for the problem of fixed solution of PDE., but of all the methods , finite difference method and finite element method are used most often.
The basic principle of finite difference method is replacing the differential quotient of each order in the equations with difference quotient of respective order. With respect to the specific problem and the difference scheme along with the proper initial and boundary conditions, we can get a linear or nonlinear algebraic equations. The chief task of difference method is constructing a reasonable difference scheme which keeps the main property of the initial problem as well is accurate enough. A good and feasible difference scheme should satisfy the consistency, stability and convergence.
In this paper, we mainly consider the following typical initial and boundary value problem of nonlinear coupling PDE.
，
，
of which
is an unknown vector function of dimensions.
is the positive definite coefficient matrix with the order, is a known vector function of dimensions , () belongs to the rectangle domain. is known vector function of dimensions.
Aiming at the fixed solution problem of above PDE, we construct the difference scheme as followings:
，，
and give the proof of stability under certain conditions.
With regard to the above fixed solution problem, if we adopt the implicit scheme, then we’ll get a nonlinear difference equations which is related to the iterative method of solving nonlinear equations. In Chapter 3 , the author give a brief introduction of the iterative method of solving nonlinear equations and with regard to the damp coefficient scheme method,
，
the author gives the proof the necessary and sufficient condition of its stability.
At the last, using the former schemes, the author give a numerical simulation of 3N concentration during the sewage disposal process. The results show that the numerical solution bears a good convergence and simulation, and consequently indicate the high practical significance of the adoptive difference scheme and the damp coefficient method in solving nonlinear equations.
人们在实践当中，对于微分方程定解问题，发展了很多方法，包括方法、各种混合元、边界元、无限元法及上世纪八十年代兴起的小波分析法，但应用最多的还是有限元法和有限差分法。
其中有限差分法的思想就是直接用各阶差商来代替微分方程中的各阶导数，根据具体问题和所选择的差分格式的不同，加上适当的初边值条件，便形成一个线性或非线性的代数方程组。差分方法的首要问题就是构造合理的差分格式，使得它的解保持原问题的某些主要性质，并且又足够精确。一个好的和实用的差分格式应满足相容性、稳定性和收敛性。
本文主要考虑一类如下形式的具有代表性的非线性耦合偏微分方程组初边值问题：
，
，
其中：
是维未知向量函数,是阶正定系数矩阵，是维向量函数，其中和属于矩形域：
是维已知
向量函数。
针对上述形式微分方程定解问题，构造如下的差分格式
，，
在适当的条件下给出稳定性的证明。
对于上述定解问题，若采用隐式差分格式，便得到一个非线性的差分方程组，这就涉及到非线性方程组数值解法问题，于是在第三章中简要介绍了非线性方程组的迭代方法，并针对如下形式的阻尼参数型迭代格式，
，
给出并证明了其收敛性的的充要条件。
最后，利用前述格式，实际应用当中我们对污水处理过程中的三氮进行了数值模拟。结果表明，数值解具有较好的收敛性和模拟效果，从而说明了所采用的差分格式和求解非线性差分方程的阻尼参数下降差分格式具有较高的实用价值。
Many mathematical models in the fields of physics, chemistry, biology and geology can be boiled down to the fixed solution problem of linear or nonlinear partial differential equations (PDE.). It’s so hard for people to get the accurate solution of nonlinear PDE as people have to switch to the numerical solution. Despite the long history of the numerical solution method of PDE, it had advance little due to the restraints of people’s computing ability. It is not until the 70s of the 20th, with the rapid development of computer, people has much strengthened his computing ability and focused the research on the nonlinear science which includes the important aspect of numerical solution of PDE.
In practice, people has developed many kinds of numerical solution methods which covers the Galerkin method and all kinds of mixed or boundary or infinite element methods as well as the wavelet method for the problem of fixed solution of PDE., but of all the methods , finite difference method and finite element method are used most often.
The basic principle of finite difference method is replacing the differential quotient of each order in the equations with difference quotient of respective order. With respect to the specific problem and the difference scheme along with the proper initial and boundary conditions, we can get a linear or nonlinear algebraic equations. The chief task of difference method is constructing a reasonable difference scheme which keeps the main property of the initial problem as well is accurate enough. A good and feasible difference scheme should satisfy the consistency, stability and convergence.
In this paper, we mainly consider the following typical initial and boundary value problem of nonlinear coupling PDE.
，
，
of which
is an unknown vector function of dimensions.
is the positive definite coefficient matrix with the order, is a known vector function of dimensions , () belongs to the rectangle domain. is known vector function of dimensions.
Aiming at the fixed solution problem of above PDE, we construct the difference scheme as followings:
，，
and give the proof of stability under certain conditions.
With regard to the above fixed solution problem, if we adopt the implicit scheme, then we’ll get a nonlinear difference equations which is related to the iterative method of solving nonlinear equations. In Chapter 3 , the author give a brief introduction of the iterative method of solving nonlinear equations and with regard to the damp coefficient scheme method,
，
the author gives the proof the necessary and sufficient condition of its stability.
At the last, using the former schemes, the author give a numerical simulation of 3N concentration during the sewage disposal process. The results show that the numerical solution bears a good convergence and simulation, and consequently indicate the high practical significance of the adoptive difference scheme and the damp coefficient method in solving nonlinear equations.
引文
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[6] 由同顺. 非线性对流扩散方程的UNO-MMOCCA差分方法. 应用数学学报,2003,26(2):264~271.
[7] 袁益让,梁栋,芮洪兴. 海水入侵防治工程的数值模拟. 计算物理,2001,18(6):556~562.
[8] 袁益让. 多层渗流方程组合系统的迎风分数步长差分方法和应用. 中国科学,A辑,2001,31(9):791~806.
[9] 袁益让. 可压缩两相驱动问题的迎风差分格式及其理论分析. 应用数学学报,2002,25(3):484~496.
[10] 由同顺. 二维对流扩散方程的非振荡特征差分方法. 计算数学,2000,22(2):159~166.
[11] 周毓麟,沈隆钧,袁光伟. 拟线性抛物组具有并行本性的无条件稳定与收敛的差分方法. 中国科学,A辑,2003,33(4):310~324.
[12] 袁光伟. 非线性抛物组非均匀网格差分解的唯一性和稳定性. 计算数学,2000,22(2):139~150.
[13] 周毓麟,沈隆钧,袁光伟.非线性抛物组具并行本性的某些实用差分格式. 数值计算与计算机应用,1997(1):64~73.
[14] 秦新强,马逸尘. 一维非线性对流占优扩散方程特征差分法的两重网格算法. 工程数学学报,2003,20(6):1~6.
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[17] 杨志峰. 高精度有限差分方法研究进展. 自然科学进展,1999,9(9):769~779.
[18] 陈玉娟. 求解一类反应找散方程组数值解的组合单调迭代法. 数学杂志,2000,20(4):452~458.
[19] 冯国胜. 沿着积分曲线求解非线性方程组. 同济大学学报,1995,23(5):577~582.
[20] 白中治. 一类求解非线性代数方程组的并行多分裂AOR算法. 应用数学和力
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[23] 廖世俊. 同伦分析方法:一种新的求解非线性问题的近似解析方法. 应用数学和力学,1998,19(10):885~890.
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[30] 李荣华,冯果枕. 微分方程数值解法(第三版). 高等教育出版社,1996.
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[2] 邬吉明,符尚武,沈隆钧. 非线性方程的一种高精度差分格式. 数值计算与计算机应用,2003(2):116~128.
[3] 奚成刚,陈家军,许兆义. 水气二相流方程的一种离散数值解法. 水科学进展,2002,13(3):287~291.
[4] 王同科. 一维对流扩散方程CRANK-NICOLSON特征差分格式. 应用数学,2001,14(4):55~60.
[5] 王同科. 二维对流扩散方程基于三角形网格的特征差分格式. 数值计算与计算机应用,2003(3):177~188.
[6] 由同顺. 非线性对流扩散方程的UNO-MMOCCA差分方法. 应用数学学报,2003,26(2):264~271.
[7] 袁益让,梁栋,芮洪兴. 海水入侵防治工程的数值模拟. 计算物理,2001,18(6):556~562.
[8] 袁益让. 多层渗流方程组合系统的迎风分数步长差分方法和应用. 中国科学,A辑,2001,31(9):791~806.
[9] 袁益让. 可压缩两相驱动问题的迎风差分格式及其理论分析. 应用数学学报,2002,25(3):484~496.
[10] 由同顺. 二维对流扩散方程的非振荡特征差分方法. 计算数学,2000,22(2):159~166.
[11] 周毓麟,沈隆钧,袁光伟. 拟线性抛物组具有并行本性的无条件稳定与收敛的差分方法. 中国科学,A辑,2003,33(4):310~324.
[12] 袁光伟. 非线性抛物组非均匀网格差分解的唯一性和稳定性. 计算数学,2000,22(2):139~150.
[13] 周毓麟,沈隆钧,袁光伟.非线性抛物组具并行本性的某些实用差分格式. 数值计算与计算机应用,1997(1):64~73.
[14] 秦新强,马逸尘. 一维非线性对流占优扩散方程特征差分法的两重网格算法. 工程数学学报,2003,20(6):1~6.
[15] 李荣华. 广义差分法及其应用. 吉林大学自然科学学报，1995(2):14~22.
[16] 锁海滨,汤渭龙,沈复. 非线性偏微分方程组的数值解在化学工程中的应用. 石油大学学报(自然科学版),1996,20(3):68~71.
[17] 杨志峰. 高精度有限差分方法研究进展. 自然科学进展,1999,9(9):769~779.
[18] 陈玉娟. 求解一类反应找散方程组数值解的组合单调迭代法. 数学杂志,2000,20(4):452~458.
[19] 冯国胜. 沿着积分曲线求解非线性方程组. 同济大学学报,1995,23(5):577~582.
[20] 白中治. 一类求解非线性代数方程组的并行多分裂AOR算法. 应用数学和力
学,1995,16(7):633~639.
[21] 孔敏,沈祖和. 解非线性方程组的极大熵方法. 高等学校计算数学学报,1999(1):1~7
[22] 蔡松柏,沈蒲生. 关于非线性方程组求解技术. 湖南大学学报(自然科学版),2000,27(3):86~91.
[23] 廖世俊. 同伦分析方法:一种新的求解非线性问题的近似解析方法. 应用数学和力学,1998,19(10):885~890.
[24] 刘停战,李光烨. 解非线性方程组的改进的换列法. 高等学校计算数学学报,1993(2):106~119.
[25] 郭本瑜. 偏微分方程的差分方法. 科学出版社,1988.
[26] 陆金甫,顾丽珍,陈景良. 偏微分方程差分方法. 高等教育出版社,1988.
[27] 李德元,陈光南. 抛物型方程差分方法引论. 科学出版社,1995.
[28] 李德元,徐国荣,水鸿寿 等. 二维非定常流体力学数值方法. 科学出版社,1987.
[29] 董光昌. 非线性二阶偏微分方程. 清华大学出版社,1988.
[30] 李荣华,冯果枕. 微分方程数值解法(第三版). 高等教育出版社,1996.
[31] 李大潜,陈韵梅. 非线性发展方程. 科学出版社,1989.
[32] 李荣华，陈仲英. 微分方程广义差分法. 吉林大学出版社，1994.
[33] 蔡大用,白峰杉. 高等数值分析. 清华大学出版社,1997.
[34] 黄象鼎,曾钟钢,马亚南. 非线性数值分析. 武汉大学出版社,2000.
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