两类偏微分方程的数值解
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摘要
本文研究两个问题．第一个问题是在记忆材料的热转导、多孔粘弹性介质的压缩、动态人口、原子反应动力学等问题中，常常碰到的抛物型积分微分方程。对于该种方程的数值求解，国外的V．Thomée，Stig．Larsson，W．Mclean，C．Lubich，J．C．López-Marcos，J．M．Sanz-Serna，G．Fairweather，L．Wahlbin，I．H．Sloan，Yanping Lin等，国内的陈传淼、黄云清、徐大、汤涛、胡齐芽、张铁等做了大量的研究，他们大多采用有限元方法，样条配置方法，有限差分方法以及谱配置方法。但本文采用的空间线性有限元，时间拉普拉斯变换数值逆却很少有人涉及。
第二个问题是炼铜堆浸工艺中的浸润面所满足的一个拟线性椭圆方程，建立不同类型堆浸控制参数及速率计算的数学模式是国内外都感兴趣的问题，尤其是在数学上准确描述出来显得特别重要而迫切。本文先用差分法离散，再用逆Broyden秩1迭代法和牛顿迭代法分别近似计算。
主要结果如下：
(1)给出一类偏积分微分方程空间线性有限元，时间拉普拉斯变换数值逆的全离散格式及数值例子。
(2)给出堆浸工艺中浸润面的非线性问题逆Broyden秩1迭代法及数值例子。
(3)给出堆浸工艺中浸润面的非线性问题牛顿迭代法及数值例子。
In this paper, we study two questions. The first is the integro-differential equation of parabolic type often occurs in applications such as heat conduction in material with memory, compression of poro-viscoelastic media, population dynamics, nuclear reactor dynamics, etc.. There are lots of documents of V. Thomée[1、5、7、16、17、18、19、20、21、22、23、24、31], Stig. Larsson [19], W. Mclean [5、17、20、24], Ch. Lubich[18], J. C. López-Marcos [14], J. M. Sanz-Serna [6], G. Fairweather[3、15], L. Wahlbin [1、17、19], I. H. Sloan [7、18、22、23], Yanping Lin [31] in overseas and Chuan-miao Chen [1、35], Yun-qing Huang [2], Da Xu [8、9、10、11、12、13], Tao Tang [33], Qiya Hu [34], Tie Zhang [38] in home. A lot of them use FEM; Spline collocation methods; finite difference methods; Spectral collocation methods. But few of them make the linear equation of the linear finite element method in the direction of of x and the inversion technique for the laplace transform in the direction of t. The second is the quasilinear equation of elliptic type in application such as infiltrates surface of heap leaching process. All are interested in building mathematical models of different types of heap leaching rates and parameters. It is particularly important and urgent to describe particularly in mathematicacs. The artical presents the difference method first to obtain the discretization, and then applies single rank inverse Broyden iterative method and newton iterative method.
Mam results follows:
(1)Given the numerical experiments and the full discretization for the linear equation of the linear finite element method in the direction of of x and the inversion technique for the laplace transform in the direction of t;
(2)Given the numerical experiments and Single Rank Inverse Broyden Iterative method for a Nonlinear Problems to Infiltrates Surface of Heap Leaching Process.
(3)Given the numerical experiments and Newton Iterative method for a Nonlinear Problems to Infiltrates Surface of Heap Leaching Process.
第二个问题是炼铜堆浸工艺中的浸润面所满足的一个拟线性椭圆方程，建立不同类型堆浸控制参数及速率计算的数学模式是国内外都感兴趣的问题，尤其是在数学上准确描述出来显得特别重要而迫切。本文先用差分法离散，再用逆Broyden秩1迭代法和牛顿迭代法分别近似计算。
主要结果如下：
(1)给出一类偏积分微分方程空间线性有限元，时间拉普拉斯变换数值逆的全离散格式及数值例子。
(2)给出堆浸工艺中浸润面的非线性问题逆Broyden秩1迭代法及数值例子。
(3)给出堆浸工艺中浸润面的非线性问题牛顿迭代法及数值例子。
In this paper, we study two questions. The first is the integro-differential equation of parabolic type often occurs in applications such as heat conduction in material with memory, compression of poro-viscoelastic media, population dynamics, nuclear reactor dynamics, etc.. There are lots of documents of V. Thomée[1、5、7、16、17、18、19、20、21、22、23、24、31], Stig. Larsson [19], W. Mclean [5、17、20、24], Ch. Lubich[18], J. C. López-Marcos [14], J. M. Sanz-Serna [6], G. Fairweather[3、15], L. Wahlbin [1、17、19], I. H. Sloan [7、18、22、23], Yanping Lin [31] in overseas and Chuan-miao Chen [1、35], Yun-qing Huang [2], Da Xu [8、9、10、11、12、13], Tao Tang [33], Qiya Hu [34], Tie Zhang [38] in home. A lot of them use FEM; Spline collocation methods; finite difference methods; Spectral collocation methods. But few of them make the linear equation of the linear finite element method in the direction of of x and the inversion technique for the laplace transform in the direction of t. The second is the quasilinear equation of elliptic type in application such as infiltrates surface of heap leaching process. All are interested in building mathematical models of different types of heap leaching rates and parameters. It is particularly important and urgent to describe particularly in mathematicacs. The artical presents the difference method first to obtain the discretization, and then applies single rank inverse Broyden iterative method and newton iterative method.
Mam results follows:
(1)Given the numerical experiments and the full discretization for the linear equation of the linear finite element method in the direction of of x and the inversion technique for the laplace transform in the direction of t;
(2)Given the numerical experiments and Single Rank Inverse Broyden Iterative method for a Nonlinear Problems to Infiltrates Surface of Heap Leaching Process.
(3)Given the numerical experiments and Newton Iterative method for a Nonlinear Problems to Infiltrates Surface of Heap Leaching Process.
引文
[1] C. Chen, V. Thomée, and L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp., Vol. 58, (1992), pp. 587-602.
[2] Huang yun-qing, Time discretization scheme for an integro-differential equation of parabolic type, J. Comp. Math., Vol. 12, 3(1994), pp. 259-263.
[3] Yi Yan, G. Fairweather, Orthogonal spline collocation methods for some partial integro-differential equations, SIAM J. Numer. Anal., Vol. 29, No. 8(1992), pp. 755-768.
[4] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., Vol. 17, (1986), pp. 704-719.
[5] W. Mclean, V. Thomée, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser., B35, (1993), pp. 23-70.
[6] J. M. Sanz-Serna, A Numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., Vol. 25, (1988), pp. 319-327.
[7] I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal., Vol. 23, (1996), pp. 1052-1061.
[8] Xu Da, The global behaviour of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO., Vol. 34, (1997), pp. 71-104.
[9] Xu Da, The long-time global behavior of time discretization for fractional order Volterra equation, CALCOLO., Vol. 35, (1998), pp. 93-116.
[10] Xu Da, Non-smooth initial data error estimates with the weight norms for the linear finite element method of parabolic partial differential equations, Appl. Math. Comp., Vol. 54, (1993), pp. 1-24.
[11] Xu Da, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel I: smooth initial data, Appl. Math. Comp., Vol. 58, (1993), pp. 1-27.
[12] Xu Da, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel II:nonsmooth initial data, Appl. Math. Comp., Vol. 58, (1993), pp. 29-60.
[13] Xu Da, Finite element methods for the nonlinear integro-differential equations, Appl. Math. Comp., Vol. 58, (1993), pp. 241-273.
[14] J. C. Lopez-Marcos, A difference scheme for a nonlinear partial integro- differential equation, SIAM J. Numer. Anal., Vol. 27, (1990), pp. 20-31.
[15] E. G. Yanik, G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis. Theory. Methods & Appl., Vol. 12, No. 8(1988), pp. 785-809.
[16] V.Thomee, Nai-Ying Zhang, Error estimates for semidiscrete finite element methods for parabolic integro-differential equation, Math. Comp., Vol. 53, (1989), pp. 121-139.
[17] W. Mclean, V. Thomee, L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comp. Appl. Math., Vol. 69, (1996), PP.49-69.
[18] C. Lubich, I. H. Sloan, V. Thomee, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp., Vol. 65, (1996), pp.1-17.
[19] Stig Larsson, V.Thomee, L. B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., Vol. 67, (1998), pp. 45-71.
[20] W. Mclean, V. Thomee, Asymptotic behavior of numerical solutions of an evotution equation with memory, Asymptotic Analysis, Vol. 14, (1997), pp. 257-276.
[21] Per-gunnar Martinsson, Discretization of certain Evolutions equation with memory using convolution quadratures, 博士论文.
[22] K. Adolfsson, M. Enelund, Stig Larsson, Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg., Vol. 192, (2003), pp. 5285-5304.
[23] D. Sheen, I. H. Sloan, V. Thome, A parallel method for time-discretization of parabolic problems based on Laplace transformation and quadrature,IMA J. Numer. Anal., Vol. 23, (2004), pp.269-299.
[24] W. Mclean, V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., Vol. 24, (2004), pp. 439-463.
[25] Chang Ho Kim and U Jin Choi, Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, J. Austral. Math. Soc. Ser. B39. Vol. 39, (1998), pp. 408-430.
[26] N. Bressan and A. Quarteroni, Analysis of Chebyshev collocation methods for parabolic equations, SIAM J. Numer. Anal., Vol. 23, (1986), pp. 1138-1153.
[27] J. A. NOHEL, K. F. SHEA, Frequency domain methods for Volterra equations, Adv. in Math., Vol. 22, (1976), No. 3, pp. 278-304.
[28] Broyden C. A Class of method for sloving nonlinear simultaneous equations Math Comp. 1965. (19): 577-593.
[29] Broyden C. A Class method for sloving nonlinear simultaous equations. Comp. J, 1969,(12):95-100.
[30] J.ortega.W Rheinboldt.Itrative solution of nonlinear equation in several variable New York:Academic Press,1970.
[31] Yanping Lin, V. Thomée, L. B. Wahlbin, Ritz-Volterra projection onto finite element spaces and application to integro-differential and Relate equations,SIAM J. Numer. Anat., Vol. 28, No. 4(1991), pp. 1047-1070.
[32] C. Bernardi and Y. Maday, Properties of some weighted Sobolev spaces and application to spectral approximations, SIAM J. Numer. Anal. Vol. 26, (1989), pp. 769-829.
[33] Tao Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., Vol. 61, (1992), pp. 373-382.
[34] Qiya Hu, Geometric meshes and their application to volterra integro-differential equation with singularities, IMA J. Numer. Anal., Vol. 18, (1998), pp. 151-164.
[35] Chuanmiao Chen, Jimin Shi, Finite Element Methods for integro-differential equations, Springer-Verlag, 1998.
[36] D.L.Jagerman,An Inversion Technique for the Laplace Transform with Application to Approximation[J].B.S.T.J. 1978,(3):669-710.
[37] 陈传淼，黄云清，有限元高精度理论，湖南科学技术出版社，1995．
[38] 张铁，发展型积分微分方程的有限元方法，东北大学出版社，2001．
[39] 陈传淼，有限元超收敛构造理论，湖南科学技术出版社，2001．
[40] 李庆杨，关治，白峰杉，数值计算原理，清华大学出版社，2003．
[41] 王瑞梅，江西铜业公司所属矿山铜堆规模探讨．铜业工程，2002(2)：5-6．
[2] Huang yun-qing, Time discretization scheme for an integro-differential equation of parabolic type, J. Comp. Math., Vol. 12, 3(1994), pp. 259-263.
[3] Yi Yan, G. Fairweather, Orthogonal spline collocation methods for some partial integro-differential equations, SIAM J. Numer. Anal., Vol. 29, No. 8(1992), pp. 755-768.
[4] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., Vol. 17, (1986), pp. 704-719.
[5] W. Mclean, V. Thomée, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser., B35, (1993), pp. 23-70.
[6] J. M. Sanz-Serna, A Numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., Vol. 25, (1988), pp. 319-327.
[7] I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal., Vol. 23, (1996), pp. 1052-1061.
[8] Xu Da, The global behaviour of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO., Vol. 34, (1997), pp. 71-104.
[9] Xu Da, The long-time global behavior of time discretization for fractional order Volterra equation, CALCOLO., Vol. 35, (1998), pp. 93-116.
[10] Xu Da, Non-smooth initial data error estimates with the weight norms for the linear finite element method of parabolic partial differential equations, Appl. Math. Comp., Vol. 54, (1993), pp. 1-24.
[11] Xu Da, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel I: smooth initial data, Appl. Math. Comp., Vol. 58, (1993), pp. 1-27.
[12] Xu Da, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel II:nonsmooth initial data, Appl. Math. Comp., Vol. 58, (1993), pp. 29-60.
[13] Xu Da, Finite element methods for the nonlinear integro-differential equations, Appl. Math. Comp., Vol. 58, (1993), pp. 241-273.
[14] J. C. Lopez-Marcos, A difference scheme for a nonlinear partial integro- differential equation, SIAM J. Numer. Anal., Vol. 27, (1990), pp. 20-31.
[15] E. G. Yanik, G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis. Theory. Methods & Appl., Vol. 12, No. 8(1988), pp. 785-809.
[16] V.Thomee, Nai-Ying Zhang, Error estimates for semidiscrete finite element methods for parabolic integro-differential equation, Math. Comp., Vol. 53, (1989), pp. 121-139.
[17] W. Mclean, V. Thomee, L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comp. Appl. Math., Vol. 69, (1996), PP.49-69.
[18] C. Lubich, I. H. Sloan, V. Thomee, Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term, Math. Comp., Vol. 65, (1996), pp.1-17.
[19] Stig Larsson, V.Thomee, L. B. Wahlbin, Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., Vol. 67, (1998), pp. 45-71.
[20] W. Mclean, V. Thomee, Asymptotic behavior of numerical solutions of an evotution equation with memory, Asymptotic Analysis, Vol. 14, (1997), pp. 257-276.
[21] Per-gunnar Martinsson, Discretization of certain Evolutions equation with memory using convolution quadratures, 博士论文.
[22] K. Adolfsson, M. Enelund, Stig Larsson, Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg., Vol. 192, (2003), pp. 5285-5304.
[23] D. Sheen, I. H. Sloan, V. Thome, A parallel method for time-discretization of parabolic problems based on Laplace transformation and quadrature,IMA J. Numer. Anal., Vol. 23, (2004), pp.269-299.
[24] W. Mclean, V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., Vol. 24, (2004), pp. 439-463.
[25] Chang Ho Kim and U Jin Choi, Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel, J. Austral. Math. Soc. Ser. B39. Vol. 39, (1998), pp. 408-430.
[26] N. Bressan and A. Quarteroni, Analysis of Chebyshev collocation methods for parabolic equations, SIAM J. Numer. Anal., Vol. 23, (1986), pp. 1138-1153.
[27] J. A. NOHEL, K. F. SHEA, Frequency domain methods for Volterra equations, Adv. in Math., Vol. 22, (1976), No. 3, pp. 278-304.
[28] Broyden C. A Class of method for sloving nonlinear simultaneous equations Math Comp. 1965. (19): 577-593.
[29] Broyden C. A Class method for sloving nonlinear simultaous equations. Comp. J, 1969,(12):95-100.
[30] J.ortega.W Rheinboldt.Itrative solution of nonlinear equation in several variable New York:Academic Press,1970.
[31] Yanping Lin, V. Thomée, L. B. Wahlbin, Ritz-Volterra projection onto finite element spaces and application to integro-differential and Relate equations,SIAM J. Numer. Anat., Vol. 28, No. 4(1991), pp. 1047-1070.
[32] C. Bernardi and Y. Maday, Properties of some weighted Sobolev spaces and application to spectral approximations, SIAM J. Numer. Anal. Vol. 26, (1989), pp. 769-829.
[33] Tao Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., Vol. 61, (1992), pp. 373-382.
[34] Qiya Hu, Geometric meshes and their application to volterra integro-differential equation with singularities, IMA J. Numer. Anal., Vol. 18, (1998), pp. 151-164.
[35] Chuanmiao Chen, Jimin Shi, Finite Element Methods for integro-differential equations, Springer-Verlag, 1998.
[36] D.L.Jagerman,An Inversion Technique for the Laplace Transform with Application to Approximation[J].B.S.T.J. 1978,(3):669-710.
[37] 陈传淼，黄云清，有限元高精度理论，湖南科学技术出版社，1995．
[38] 张铁，发展型积分微分方程的有限元方法，东北大学出版社，2001．
[39] 陈传淼，有限元超收敛构造理论，湖南科学技术出版社，2001．
[40] 李庆杨，关治，白峰杉，数值计算原理，清华大学出版社，2003．
[41] 王瑞梅，江西铜业公司所属矿山铜堆规模探讨．铜业工程，2002(2)：5-6．