一类偏积分微分方程的数值计算
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摘要
在记忆材料的热传导、多孔粘弹性介质的压缩、原子反应、动力学等问题中，常常碰到抛物型偏积分微分方程，对于该方程的数值求解，国外的V.thomee([1、2、8、11、12、14])，W.Mclean([2、8、14])，Ch.Lubich([15、16])，L.Wahlbin([1])，Sanz-Serna([3])，E.G.Yanik，G.Fairweather([5])，国内的陈传淼([1])、黄元清([10])、徐大([17、18])等做了大量的研究，他们采用了有限元方法、谱配置方法及样条配置方法，但用六点隐格、拉普拉斯变换数值逆、Lubich的拉普拉斯变换数值逆离散却很少涉及。
本文考虑一类带弱奇异核抛物型偏积分微分方程时间、空间全离散格式，采用六点隐格式和拉普拉斯变换数值逆离散等方法进行数值计算，主要结果如下：
(1)给出偏积分微分方程空间x方向用六点隐格式离散，时间t方向用拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
(2)给出偏积分微分方程空间x方向用六点隐格式离散，时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
(3)给出常微分方程用Lubich的拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
以上几种方法计算结果精度较高，并且计算也比较简便。
The integro-differential equation of parabolic type oftenoccurs in application such as heat conduction in material withmemory, comression of poro-viscoelastic media, nuclear reactordynamics,ect. theres are lots of documents of V. thomee ([1、2、8、11、12、14]),W. Mclean ([2、8、14]), Ch. Lubich ([15、16]),L. Wahlbin([1]),Sanz-Serna([3]), E.G. Yanik, G. Fairwea-ther ([5]) in overseas and Chuan-miao Chen ([1])、Yuan-qingHuang ([10]),Da-Xu ([17、18]) in home. A lot of them useFEM;Spectral collocation methods;Spline collocation methods.
but few of them use six-point implicit scheme、numericalinversion for the laplace transform、umerical inversion forthe laplace transform of Lubich.
We study a partial integro-differential equations of para-bolic type with a weakly singular kernel, which use six-pointimplicit scheme and numerical inversion for the laplace tran-sform fully discrete in time and space for numerical calcul-ation, main results follows:
(1) Give a kind of fully discrete scheme of a partialintegro-differential equations which use six-point implicitscheme in the direction of space x and numerical inversion forthe laplace transform in the direction of time t for numericalcalculation.
(2) Give a kind of fully discrete scheme of a partialintegro-differential equations which use six-point implicitscheme in the direction of space x and numerical inversion forthe laplace transform of Lubich in the direction of time t for numerical calculation.
(3) Give a kind of fully discrete scheme for constantdifferential equations which use numerical inversion for thelaplace transform of Lubich for numerical calculation.
Calculating result of some methods is accuratly higher,and calculate is simpler also.
本文考虑一类带弱奇异核抛物型偏积分微分方程时间、空间全离散格式，采用六点隐格式和拉普拉斯变换数值逆离散等方法进行数值计算，主要结果如下：
(1)给出偏积分微分方程空间x方向用六点隐格式离散，时间t方向用拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
(2)给出偏积分微分方程空间x方向用六点隐格式离散，时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
(3)给出常微分方程用Lubich的拉普拉斯变换数值逆离散的全离散格式，并进行数值计算。
以上几种方法计算结果精度较高，并且计算也比较简便。
The integro-differential equation of parabolic type oftenoccurs in application such as heat conduction in material withmemory, comression of poro-viscoelastic media, nuclear reactordynamics,ect. theres are lots of documents of V. thomee ([1、2、8、11、12、14]),W. Mclean ([2、8、14]), Ch. Lubich ([15、16]),L. Wahlbin([1]),Sanz-Serna([3]), E.G. Yanik, G. Fairwea-ther ([5]) in overseas and Chuan-miao Chen ([1])、Yuan-qingHuang ([10]),Da-Xu ([17、18]) in home. A lot of them useFEM;Spectral collocation methods;Spline collocation methods.
but few of them use six-point implicit scheme、numericalinversion for the laplace transform、umerical inversion forthe laplace transform of Lubich.
We study a partial integro-differential equations of para-bolic type with a weakly singular kernel, which use six-pointimplicit scheme and numerical inversion for the laplace tran-sform fully discrete in time and space for numerical calcul-ation, main results follows:
(1) Give a kind of fully discrete scheme of a partialintegro-differential equations which use six-point implicitscheme in the direction of space x and numerical inversion forthe laplace transform in the direction of time t for numericalcalculation.
(2) Give a kind of fully discrete scheme of a partialintegro-differential equations which use six-point implicitscheme in the direction of space x and numerical inversion forthe laplace transform of Lubich in the direction of time t for numerical calculation.
(3) Give a kind of fully discrete scheme for constantdifferential equations which use numerical inversion for thelaplace transform of Lubich for numerical calculation.
Calculating result of some methods is accuratly higher,and calculate is simpler also.
引文
[1] C. Chen, V. thomee, and L. B. Wahlbin. Finite element approximation of a parabolic integro-differential equations with a weakly singular kernel. Math. Comp., 1992, Vol.58: pp. 587-602.
[2] W. Mclean, V. thomee. Numerical solution of an evolution equation with a positive type memory term. J. Austral. Math. Soc. Ser., 1993, B35: pp. 23-70.
[3] J. M. Sane-Serna. A Numerical method for a partial integro-differential equations. SIAM. J. Numer. Anal., 1998, Vol. 25:pp. 319-327.
[4] Xu Da. The global behaviour of time discretization for an abstract Volterra equation In Hilbert space. CALCOLO, 1997, Vol. 34:pp. 71-104.
[5] E. G. Yanik, G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis. Theory.Methods & Appl. ,1998 Vol. 12 (8):pp. 785-809.
[6] YiYan, G. Fairweather. Orthogonal spline collocation methods for some partial integro-differential equations. SIAM. J. Numer. Anal, 1992,Vol.29(8): pp.755-768.
[7] K. Adolfsson, M. Enelund. Stig Larsson Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel. Comput. Methods Appl. Mech. Engrg., 2003,Vol. 192:pp. 5285-5304.
[8] W. Mclean, V. thomee. Time disretization of an evolution equation via Laplace transforms. IMA. J. Numer. Anal., 2004, Vol. 24:pp. 439-463.
[9] Tao tang. Superconvergence of numerical solution to weakly singular Volerra integro-differential equations. Numer. Math., 1992, Vol. 61:pp. 373-382.
[10] Huangyun-qing. Time discretization scheme for an integro-differential equation of parabolic type. J. Comp. Math., 1994, Vol. 12(3): pp. 259-263.
[11] D. L. Jagerman. An Inversion Technique for the Laplace Transform with Application to Approximation. B. S. T. J., 1978, 57, No. 3:pp. 669-710.
[12] D. L. Jagerman. An Inversion Technique for the Laplace Transform, B. S. T. J., 1982,61, No. 8:pp. 1995-2002.
[13] Zhi-Zhong Sun. An Unconditionally Stable and 0(τ~2+h~4) Order L∞ Convergent Difference Scheme for Linear Parabolic Equations with Variable Coefficients. Numer Methods Partial Differential Eq.,2001, Vol. 17:pp. 619-631.
[14] V. Thomee and W. Mclean. Time Discretization of an Evolution Equation via Laplace Transforms. AMR, 2003, No. 7:pp.1-27.
[15] C. Lubich. Convolution Quadrature and Discretized Operational Calculus Ⅰ. Numer. Math.,1998, Vol. 52:pp. 129-145.
[16] C. Lubich. Convolution Quadrature and Discretized Operational Calculus Ⅱ. Numer. Math.,1998, Vol. 52: pp. 412-425.
[17] Xu-Da. On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅰ: Smooth Initial Data. Appl. Math. Comp.,1993, Vol. 58:pp.1-27.
[18] Xu-Da. On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅱ: Non Smooth Initial Data. Appl. Math. Comp.,1993, Vol. 58:pp. 29-61.
[19] 杨晓霖，徐大．拉普拉斯变换的数字逆在微分方程中的应用[J]．湖南师范大学自然科学学报，2004，Vol．27(2)：pp．21-25．
[20] 杨晓霖，徐大．拉普拉斯变换的数字逆在偏微分方程中的应用[J]．南华大学学报，2005，Vol．19(2)：pp．10-13．
[21] 陆金甫，关治．偏微分方程数值解法[M]．第二版．北京：清华大学出版社，2003．
[22] 曾金平．数值计算方法[M]．湖南：湖南大学出版社，2004．
[23] 钟玉泉．复变函数论[M]．第二版．北京：高等教育出版社，1987．
[2] W. Mclean, V. thomee. Numerical solution of an evolution equation with a positive type memory term. J. Austral. Math. Soc. Ser., 1993, B35: pp. 23-70.
[3] J. M. Sane-Serna. A Numerical method for a partial integro-differential equations. SIAM. J. Numer. Anal., 1998, Vol. 25:pp. 319-327.
[4] Xu Da. The global behaviour of time discretization for an abstract Volterra equation In Hilbert space. CALCOLO, 1997, Vol. 34:pp. 71-104.
[5] E. G. Yanik, G. Fairweather. Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis. Theory.Methods & Appl. ,1998 Vol. 12 (8):pp. 785-809.
[6] YiYan, G. Fairweather. Orthogonal spline collocation methods for some partial integro-differential equations. SIAM. J. Numer. Anal, 1992,Vol.29(8): pp.755-768.
[7] K. Adolfsson, M. Enelund. Stig Larsson Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel. Comput. Methods Appl. Mech. Engrg., 2003,Vol. 192:pp. 5285-5304.
[8] W. Mclean, V. thomee. Time disretization of an evolution equation via Laplace transforms. IMA. J. Numer. Anal., 2004, Vol. 24:pp. 439-463.
[9] Tao tang. Superconvergence of numerical solution to weakly singular Volerra integro-differential equations. Numer. Math., 1992, Vol. 61:pp. 373-382.
[10] Huangyun-qing. Time discretization scheme for an integro-differential equation of parabolic type. J. Comp. Math., 1994, Vol. 12(3): pp. 259-263.
[11] D. L. Jagerman. An Inversion Technique for the Laplace Transform with Application to Approximation. B. S. T. J., 1978, 57, No. 3:pp. 669-710.
[12] D. L. Jagerman. An Inversion Technique for the Laplace Transform, B. S. T. J., 1982,61, No. 8:pp. 1995-2002.
[13] Zhi-Zhong Sun. An Unconditionally Stable and 0(τ~2+h~4) Order L∞ Convergent Difference Scheme for Linear Parabolic Equations with Variable Coefficients. Numer Methods Partial Differential Eq.,2001, Vol. 17:pp. 619-631.
[14] V. Thomee and W. Mclean. Time Discretization of an Evolution Equation via Laplace Transforms. AMR, 2003, No. 7:pp.1-27.
[15] C. Lubich. Convolution Quadrature and Discretized Operational Calculus Ⅰ. Numer. Math.,1998, Vol. 52:pp. 129-145.
[16] C. Lubich. Convolution Quadrature and Discretized Operational Calculus Ⅱ. Numer. Math.,1998, Vol. 52: pp. 412-425.
[17] Xu-Da. On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅰ: Smooth Initial Data. Appl. Math. Comp.,1993, Vol. 58:pp.1-27.
[18] Xu-Da. On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅱ: Non Smooth Initial Data. Appl. Math. Comp.,1993, Vol. 58:pp. 29-61.
[19] 杨晓霖，徐大．拉普拉斯变换的数字逆在微分方程中的应用[J]．湖南师范大学自然科学学报，2004，Vol．27(2)：pp．21-25．
[20] 杨晓霖，徐大．拉普拉斯变换的数字逆在偏微分方程中的应用[J]．南华大学学报，2005，Vol．19(2)：pp．10-13．
[21] 陆金甫，关治．偏微分方程数值解法[M]．第二版．北京：清华大学出版社，2003．
[22] 曾金平．数值计算方法[M]．湖南：湖南大学出版社，2004．
[23] 钟玉泉．复变函数论[M]．第二版．北京：高等教育出版社，1987．