基于积分过程的Chebyshev-Tau高精度方法求解刚性微分方程
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摘要
刚性微分方程描述了相互作用但变化速度相差悬殊的物理或化学过程,这一刚性现象使得采用传统的微分方程数值积分方法求解遇到困难.为了实现刚性微分方程的高精度数值计算,提出了一种基于积分过程的Chebyshev-Tau方法.该方法利用了Chebyshev多项式的不定积分公式,并且采用矩阵和向量的运算形式得以实现.数值实验结果表明基于积分过程的Chebyshev-Tau方法离散一维问题得到的系数矩阵是良态的,条件数不随多项式展开阶次的提高而增长.对线性和非线性刚性微分方程的求解均实现了指数阶收敛精度.与一些经典的数值方法相比,基于积分过程的Chebyshev-Tau方法耗费较小的计算代价得到了更高的精度.
Stiff differential equations describe physical or chemical processes which interact and have quite different changing speeds. This stiff phenomenon leads the classical numerical integration methods of ordinary differential equations to have great difficulties. The Chebyshev-Tau method based on the integration process is proposed to solve stiff differential equations with high-order accuracy. This method uses the indefinite integral formula of Chebyshev polynomial,and is implemented by matrix and vector operations. Numerical results show that the coefficient matrix of the Chebyshev-Tau method based on the integration process is well-conditioned for one-dimensional problems,that is,the condition number does not increase with the increasing order of polynomial expansions. The accuracy achieves exponential convergence for both linear and nonlinear stiff differential equations. Compared with some classical numerical methods,the Chebyshev-Tau method based on the integration process obtains higher accuracy with much less computation cost.
Stiff differential equations describe physical or chemical processes which interact and have quite different changing speeds. This stiff phenomenon leads the classical numerical integration methods of ordinary differential equations to have great difficulties. The Chebyshev-Tau method based on the integration process is proposed to solve stiff differential equations with high-order accuracy. This method uses the indefinite integral formula of Chebyshev polynomial,and is implemented by matrix and vector operations. Numerical results show that the coefficient matrix of the Chebyshev-Tau method based on the integration process is well-conditioned for one-dimensional problems,that is,the condition number does not increase with the increasing order of polynomial expansions. The accuracy achieves exponential convergence for both linear and nonlinear stiff differential equations. Compared with some classical numerical methods,the Chebyshev-Tau method based on the integration process obtains higher accuracy with much less computation cost.
引文
[1]张国凤,孙红蕊.一些具有较少迭代次数和较好稳定性的新型PDIRK方法[J].兰州大学学报(自然科学版),2003,39(5):1-5.
[2]陈蓉.求解刚性振荡问题的Rosenbrock方法[D].湘潭:湘潭大学,2011.
[3]王冰,海云莎.用车贝雪夫多项式作中国西南雨季降水场预报[J].云南大学学报(自然科学版),2001,23(4):294-296.
[4]李县法,何科荣,李华.大亚湾海域潮流场谱方法数值模拟[J].科学技术与工程,2007,7(9):1847-1853.
[5]郑明,曹德贤,邹石磊,等.波动方程的第二类Chebyshev小波数值解[J].云南民族大学学报(自然科学版),2017,26(4):292-295.
[6]王建瑜,宋菲.二维Poisson方程的谱方法求解[J].科学技术与工程,2009,9(12):3425-3428.
[7] KONG Wei-bin,WU Xiong-hua. Chebyshev tau matrix method for Poisson-type equations in irregular domain[J]. Journal of Computational and Applied Mathematics,2009,228(1):158-167.
[8] MAI DUY N. An effective spectral collocation method for the direct solution of high-order ODEs[J]. Communications in Numerical Methods in Engineering,2006,22(6):627-642.
[9] WANG Li-lian,SAMSON M D,ZHAO Xiao-dan. A well-conditioned collocation method using a pseudospectral integration matrix[J]. Siam Journal on Scientific Computing,2014,36(3):A907-A929.
[10] BOYD J P. Chebyshev and Fourier spectral methods[M]. New York:Dover publications,2001:497-498.
[11]同济大学计算数学教研室.现代数值数学和计算[M].上海:同济大学出版社,2003:255-256.
[12] FRANCO J M,GOMEZ I,RANDEZ L. SDIRK methods for stiff ODEs with oscillating solutions[J]. Journal of Computational and Applied Mathematics,1997,81(2):197-209.
[13] GRASMAN J,VELING E J M,WILLEMS G M. Relaxation oscillations governed by a Van Der Pol equation with periodic forcing term[J]. Siam Journal on Applied Mathematics,1976,31(4):667-676.
[2]陈蓉.求解刚性振荡问题的Rosenbrock方法[D].湘潭:湘潭大学,2011.
[3]王冰,海云莎.用车贝雪夫多项式作中国西南雨季降水场预报[J].云南大学学报(自然科学版),2001,23(4):294-296.
[4]李县法,何科荣,李华.大亚湾海域潮流场谱方法数值模拟[J].科学技术与工程,2007,7(9):1847-1853.
[5]郑明,曹德贤,邹石磊,等.波动方程的第二类Chebyshev小波数值解[J].云南民族大学学报(自然科学版),2017,26(4):292-295.
[6]王建瑜,宋菲.二维Poisson方程的谱方法求解[J].科学技术与工程,2009,9(12):3425-3428.
[7] KONG Wei-bin,WU Xiong-hua. Chebyshev tau matrix method for Poisson-type equations in irregular domain[J]. Journal of Computational and Applied Mathematics,2009,228(1):158-167.
[8] MAI DUY N. An effective spectral collocation method for the direct solution of high-order ODEs[J]. Communications in Numerical Methods in Engineering,2006,22(6):627-642.
[9] WANG Li-lian,SAMSON M D,ZHAO Xiao-dan. A well-conditioned collocation method using a pseudospectral integration matrix[J]. Siam Journal on Scientific Computing,2014,36(3):A907-A929.
[10] BOYD J P. Chebyshev and Fourier spectral methods[M]. New York:Dover publications,2001:497-498.
[11]同济大学计算数学教研室.现代数值数学和计算[M].上海:同济大学出版社,2003:255-256.
[12] FRANCO J M,GOMEZ I,RANDEZ L. SDIRK methods for stiff ODEs with oscillating solutions[J]. Journal of Computational and Applied Mathematics,1997,81(2):197-209.
[13] GRASMAN J,VELING E J M,WILLEMS G M. Relaxation oscillations governed by a Van Der Pol equation with periodic forcing term[J]. Siam Journal on Applied Mathematics,1976,31(4):667-676.