非线性抛物型方程计算方法
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摘要
本文简要回顾非线性抛物型方程差分方法若干研究工作,包括周毓麟先生在该研究方向取得的部分研究成果,并对近年来相关的部分研究进展进行综述,展望拟开展的研究工作.
In this paper, we briefly review some work on finite difference methods for nonlinear parabolic equations, including parts of achievements contributed by Professor ZHOU Yulin. Moreover, a survey of recent advances on discretization methods for nonlinear parabolic equations is presented and some plans on future research are described.
In this paper, we briefly review some work on finite difference methods for nonlinear parabolic equations, including parts of achievements contributed by Professor ZHOU Yulin. Moreover, a survey of recent advances on discretization methods for nonlinear parabolic equations is presented and some plans on future research are described.
引文
1Zhou Y L.Applications of Discrete Functional Analysis to Finite Difference Method.Beijing:International Academic Publishers,1990
2周毓麟.周毓麟论文集(第一卷).北京:科学出版社,1992
3周毓麟.周毓麟论文集(第二卷,第三卷).北京:世界图书出版公司,2002
4Heinrich B.Finite Difference Methods on Irregular Networks.International Series of Numerical Mathematics,82.Basel-Boston-Stuttgart:Birkhauser-Verlag,1987
5李荣华,陈仲英.微分方程广义差分法.长春:吉林大学出版社,1994(Li R H,Chen Z Y,Wu W.Generalized Difference Methods for Differential Equations:Numerical Analysis of Finite Volume Methods.Monographs and Textbooks in Pure and Applied Mathematics,226.New York:Marcel Dekker,2000)
6Shashkov M.Conservative Finite Difference Methods on General Grids.Boca Raton:CRC Press,1996
7Eymard R,Gallou¨et T,Herbin R.Finite volume methods.In:Ciarlet P G,Lions J L,eds.Handbook of Numerical Analysis,vol.VII.Amsterdam:North-Holland,2000,713–1020
8李德元,陈光南.抛物型方程差分方法引论.北京:科学出版社,1995
9郭本瑜.偏微分方程差分方法.北京:科学出版社,1988
10萨马尔斯基A A,安德烈耶夫B B.椭圆型方程差分方法.武汉大学计算数学教研室,译.北京:科学出版社,1984
11Marchuk G I,Shaidurov V V.Difference Methods and Their Extrapolations.Berlin:Springer-Verlag,1983
12Morton K M.Numerical Solution of Convection-Diffusion Problems.London:Chapman&Hall,1996
13Zhou Y L.Interpolation formulas of intermediate quotients of discrete functions with several indices.J Comput Math,1984,2:376–381
14Zhou Y L.Interpolation formulas of intermediate quotients of discrete functions and their applications.In:Feng K,ed.Proceedings of the1984Beijing Symposium on Differential Geometry and Differential Equations.Finite Difference Method and Finite Slice Method for Nonlinear Evolutional Systems of Partial Differential Equations.1984,126–152
15Zhou Y L.General interpolation formulas for spaces of discrete functions with one index.Acta Sci Nat Univ Pekinensis,1992,28:1–16
16Zhou Y L.On the general interpolation fromulas for spaces of discrete functions(I).J Comput Math,1993,11:188–192
17Zhou Y L.General interpolation formulas for spaces of discrete functions with nonuniform meshes.J Comput Math,1995,13:70–93
18Zhou Y L,Yuan G W.Some discrete interpolation inequalities with several indices.Beijing Math,1996,2:94–108
19Glitzky A,Griepentrog J A.Discrete Sobolev-Poincar`e inequalities for Vorono¨finite volume approximations.SIAM J Numer Anal,2010,48:372–391
20Marianne B C,Claire C H,Francis F.On discrete functional inequalities for some finite volume schemes.arXiv:1202.4860,2012
21周毓麟.拟线性抛物方程组第一边界问题的有限差分法.中国科学A辑,1985,15:206–220(Zhou Y L.Finite difference method of first boundary problem for quasilinear parabolic systems.Sci China Ser A,1985,28:368–385)
22周毓麟,沈隆钧,韩臻.拟线性抛物组第一边界问题的有限差分法(续).中国科学A辑,1990,20:1126–1136(Zhou Y L,Shen L J,Han Z.Finite difference method of first boundary problem for quasilinear parabolic systems(continued).Sci China Ser A,1991,34:405–418)
23周毓麟,沈隆钧,袁光伟.拟线性抛物组第一边值问题有限差分方法III:稳定性.中国科学A辑,1996,26:577–583(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,III:Stability.Sci China Ser A,1996,39:685–693)
24周毓麟,沈隆钧,袁光伟.拟线性抛物方程组第一边界问题的有限差分方法IV:迭代收敛性.中国科学A辑,1997,27:322–327(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,IV:Convergence of iteration.Sci China Ser A,1997,40:469–474)
25周毓麟,沈隆钧,袁光伟.拟线性抛物方程组第一边界问题的有限差分方法V:迭代差分格式的收敛性.中国科学A辑,1997,27:988–996(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,V:Convergence of iterative difference schemes.Sci China Ser A,1997,40:1148–1157)
26Zhou Y L,Shen L J,Yuan G W.Convergence of iterative difference schemes for two and three dimensional nonlinear parabolic systems.J Partial Differential Equations,1997,10:257–274
27周毓麟,袁光伟.两维完全非线性伪抛物组的差分格式.中国科学A辑,1995,25:1248–1258(Zhou Y L,Yuan G W.Difference schemes for fully nonlinmear pseudo-parabolic systems with two space dimensions.Sci China Ser A,1996,39:122–136)
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29Zhou Y L.Finite difference method of boundary problem for nonlinear pseudo-hyperbolic systems.J Math Res Expos,1985,5:73–79
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31Zhou Y L,Guo B L.Finite difference solutions of the boundary problems for the systems of ferro-magnetic chain.J Comput Math,1983,1:294–302
32Zhou Y L,Xu G R.Finite difference solutions of the nonlinear murual boundary problems for the systems of ferro-magnetic chain.J Comput Math,1984,2:263–271
33周毓麟,袁国兴.非线性伪双曲差分格式的收敛性.清华大学学报(自然科学版),1988,28:10–17
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99Yue J,Yuan G.Picard-Newton iterative method with time step control for multi-material non-equilibrium radiation diffusion problem.Commun Comput Phys,2011,10:844–866
100袁光伟,沈隆钧.抛物型方程实验的隐式区域分解方法(I).计算物理实验室年报,2000,472–489
101袁光伟,沈隆钧.抛物型方程实验的隐式区域分解方法(II).计算物理实验室年报,2000,490–499
102袁光伟,杭旭登.热传导方程基于界面修正的迭代并行计算方法.计算物理实验室年报,2003,535–549
103Sheng Z,Yuan G,Hang X.Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation.Appl Math Comput,2007,184:1015–1031
104袁光伟,杭旭登,盛志强.拟线性抛物方程组具有界面外推的并行本性差分方法.中国科学A辑,2007,37:145–164(Yuan G,Hang X,Sheng Z.Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems.Sci China Ser A,2007,50:253–275)
105Yuan G,Sheng Z,Hang X.The unconditional stability of parallel difference schemes with second order convergence for nonlinear parabolic system.J Partial Differential Equations,2007,20:45–64
106Shi H,Liao H.Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations.SIAM J Numer Anal,2006,44:1584–1611
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108Yuan G,Yao Y,Yin L.Conservative domain decomposition procedure for nonlinear diffusion problems on arbitrary quadrilateral grids.SIAM J Sci Comput,2011,33:1352–1368
2周毓麟.周毓麟论文集(第一卷).北京:科学出版社,1992
3周毓麟.周毓麟论文集(第二卷,第三卷).北京:世界图书出版公司,2002
4Heinrich B.Finite Difference Methods on Irregular Networks.International Series of Numerical Mathematics,82.Basel-Boston-Stuttgart:Birkhauser-Verlag,1987
5李荣华,陈仲英.微分方程广义差分法.长春:吉林大学出版社,1994(Li R H,Chen Z Y,Wu W.Generalized Difference Methods for Differential Equations:Numerical Analysis of Finite Volume Methods.Monographs and Textbooks in Pure and Applied Mathematics,226.New York:Marcel Dekker,2000)
6Shashkov M.Conservative Finite Difference Methods on General Grids.Boca Raton:CRC Press,1996
7Eymard R,Gallou¨et T,Herbin R.Finite volume methods.In:Ciarlet P G,Lions J L,eds.Handbook of Numerical Analysis,vol.VII.Amsterdam:North-Holland,2000,713–1020
8李德元,陈光南.抛物型方程差分方法引论.北京:科学出版社,1995
9郭本瑜.偏微分方程差分方法.北京:科学出版社,1988
10萨马尔斯基A A,安德烈耶夫B B.椭圆型方程差分方法.武汉大学计算数学教研室,译.北京:科学出版社,1984
11Marchuk G I,Shaidurov V V.Difference Methods and Their Extrapolations.Berlin:Springer-Verlag,1983
12Morton K M.Numerical Solution of Convection-Diffusion Problems.London:Chapman&Hall,1996
13Zhou Y L.Interpolation formulas of intermediate quotients of discrete functions with several indices.J Comput Math,1984,2:376–381
14Zhou Y L.Interpolation formulas of intermediate quotients of discrete functions and their applications.In:Feng K,ed.Proceedings of the1984Beijing Symposium on Differential Geometry and Differential Equations.Finite Difference Method and Finite Slice Method for Nonlinear Evolutional Systems of Partial Differential Equations.1984,126–152
15Zhou Y L.General interpolation formulas for spaces of discrete functions with one index.Acta Sci Nat Univ Pekinensis,1992,28:1–16
16Zhou Y L.On the general interpolation fromulas for spaces of discrete functions(I).J Comput Math,1993,11:188–192
17Zhou Y L.General interpolation formulas for spaces of discrete functions with nonuniform meshes.J Comput Math,1995,13:70–93
18Zhou Y L,Yuan G W.Some discrete interpolation inequalities with several indices.Beijing Math,1996,2:94–108
19Glitzky A,Griepentrog J A.Discrete Sobolev-Poincar`e inequalities for Vorono¨finite volume approximations.SIAM J Numer Anal,2010,48:372–391
20Marianne B C,Claire C H,Francis F.On discrete functional inequalities for some finite volume schemes.arXiv:1202.4860,2012
21周毓麟.拟线性抛物方程组第一边界问题的有限差分法.中国科学A辑,1985,15:206–220(Zhou Y L.Finite difference method of first boundary problem for quasilinear parabolic systems.Sci China Ser A,1985,28:368–385)
22周毓麟,沈隆钧,韩臻.拟线性抛物组第一边界问题的有限差分法(续).中国科学A辑,1990,20:1126–1136(Zhou Y L,Shen L J,Han Z.Finite difference method of first boundary problem for quasilinear parabolic systems(continued).Sci China Ser A,1991,34:405–418)
23周毓麟,沈隆钧,袁光伟.拟线性抛物组第一边值问题有限差分方法III:稳定性.中国科学A辑,1996,26:577–583(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,III:Stability.Sci China Ser A,1996,39:685–693)
24周毓麟,沈隆钧,袁光伟.拟线性抛物方程组第一边界问题的有限差分方法IV:迭代收敛性.中国科学A辑,1997,27:322–327(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,IV:Convergence of iteration.Sci China Ser A,1997,40:469–474)
25周毓麟,沈隆钧,袁光伟.拟线性抛物方程组第一边界问题的有限差分方法V:迭代差分格式的收敛性.中国科学A辑,1997,27:988–996(Zhou Y L,Shen L J,Yuan G W.Finite difference method of first boundary problem for quasilinear parabolic systems,V:Convergence of iterative difference schemes.Sci China Ser A,1997,40:1148–1157)
26Zhou Y L,Shen L J,Yuan G W.Convergence of iterative difference schemes for two and three dimensional nonlinear parabolic systems.J Partial Differential Equations,1997,10:257–274
27周毓麟,袁光伟.两维完全非线性伪抛物组的差分格式.中国科学A辑,1995,25:1248–1258(Zhou Y L,Yuan G W.Difference schemes for fully nonlinmear pseudo-parabolic systems with two space dimensions.Sci China Ser A,1996,39:122–136)
28Zhou Y L.Finite difference method of the boundary problem for the systems of Sobolev-Galpern type.J Comput Math,1983,1:363–371
29Zhou Y L.Finite difference method of boundary problem for nonlinear pseudo-hyperbolic systems.J Math Res Expos,1985,5:73–79
30Zhou Y L.The general nonlinear mutual boundary problems for the systems of nonlinear wave equations by finite difference method.J Comput Math,1985,3:134–160
31Zhou Y L,Guo B L.Finite difference solutions of the boundary problems for the systems of ferro-magnetic chain.J Comput Math,1983,1:294–302
32Zhou Y L,Xu G R.Finite difference solutions of the nonlinear murual boundary problems for the systems of ferro-magnetic chain.J Comput Math,1984,2:263–271
33周毓麟,袁国兴.非线性伪双曲差分格式的收敛性.清华大学学报(自然科学版),1988,28:10–17
34Zhou Y L,Du M S.Difference schemes for fully nonlinear pseudo-hyperbolic systems.Acta Math Appl Sin(Engl Ser),1991,7:298–320
35Zhou Y L.Difference schemes with nonuniform meshes for nonlinear parabolic system.J Comput Math,1996,14:319–335
36Zhou Y L.Finite difference method with nonuniform meshes for quasilinear parabolic systems.J Comput Math,1997,15:289–310
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