ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

doi:10.1007/s10473-019-0405-1

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (4): 971-988.doi: 10.1007/s10473-019-0405-1

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

Temur JANGVELADZE^1,2, Zurab KIGURADZE^1,3, Mikheil GAGOSHIDZE^1,2

1. Ilia Vekua Institute of Applied Mathematics, Ivane Javakhishvili Tbilisi State University, 2 University St. 0186, Tbilisi, Georgia;
2. Georgian Technical University, 77 Kostava Ave. 0175, Tbilisi, Georgia;
3. Missouri University Science and Technology, Department of Electrical and Computer Engineering, Electromagnetic Compatibility Laboratory, 4000 Enterprise Drive, 65409 Rolla, MO, USA

收稿日期:2017-12-12 出版日期:2019-08-25 发布日期:2019-09-12
作者简介:Temur JANGVELADZE,E-mail:temur.jangveladze@tsu.ge,t.jangveladze@gtu.ge,tjangv@yahoo.com;Zurab KIGURADZE,E-mail:kiguradzz@mst.edu,zkigur@yahoo.com;Mikheil GAGOSHIDZE,E-mail:mikheil.gagoshidze@tsu.ge,m.gagoshidze@gtu.ge,mishagagoshidze@gmail.com

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

Temur JANGVELADZE^1,2, Zurab KIGURADZE^1,3, Mikheil GAGOSHIDZE^1,2

1. Ilia Vekua Institute of Applied Mathematics, Ivane Javakhishvili Tbilisi State University, 2 University St. 0186, Tbilisi, Georgia;
2. Georgian Technical University, 77 Kostava Ave. 0175, Tbilisi, Georgia;
3. Missouri University Science and Technology, Department of Electrical and Computer Engineering, Electromagnetic Compatibility Laboratory, 4000 Enterprise Drive, 65409 Rolla, MO, USA

Received:2017-12-12 Online:2019-08-25 Published:2019-09-12

摘要/Abstract

摘要： The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.

关键词: System of nonlinear partial differential equations, variable directions finite difference scheme, stability and convergence, numerical resolution

Abstract: The multi-dimensional system of nonlinear partial differential equations is considered. In two-dimensional case, this system describes process of vein formation in higher plants. Variable directions finite difference scheme is constructed. The stability and convergence of that scheme are studied. Numerical experiments are carried out. The appropriate graphical illustrations and tables are given.

Key words: System of nonlinear partial differential equations, variable directions finite difference scheme, stability and convergence, numerical resolution

中图分类号:

65M06

Temur JANGVELADZE, Zurab KIGURADZE, Mikheil GAGOSHIDZE. ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM[J]. 数学物理学报(英文版), 2019, 39(4): 971-988.

Temur JANGVELADZE, Zurab KIGURADZE, Mikheil GAGOSHIDZE. ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM[J]. Acta mathematica scientia,Series B, 2019, 39(4): 971-988.

参考文献

[1] Mitchison G J. A model for vein formation in higher plants. Proc R Soc Lond B, 1980, 207:79-109
[2] Bell J, Cosner C, Bertiger W. Solutions for a flux-dependent diffusion model. SIAM J Math Anal, 1982, 13:758-769
[3] Roussel C J, Roussel M R. Reaction-diffusion models of development with state-dependent chemical diffusion coefficients. Prog Biophys Mol Biol, 2004, 86:113-160
[4] Dzhangveladze T A. Averaged model of sum approximation for a system of nonlinear partial differential equations (Russian). Tbiliss Gos Univ Inst Prikl Mat Trudy (Proc I Vekua Inst Appl Math), 1987, 19:60-73
[5] Dzhangveladze T A, Tagvarelia T G, Convergence of a difference scheme for a system of nonlinear partial differential equations, that arise in biology (Russian). Tbiliss Gos Univ Inst Prikl Mat Trudy (Proc I Vekua Inst Appl Math), 1990, 40:77-83
[6] Jangveladze T A. The difference scheme of the type of variable directions for one system of nonlinear partial differential equations. Proc I Vekua Inst Appl Math, 1992, 42:45-66
[7] Jangveladze T, Nikolishvili M, Tabatadze B. On one nonlinear two-dimensional diffusion system. Proc 15th WSEAS Int Conf Applied Math (MATH 10), 2010:105-108
[8] Jangveladze T, Kiguradze Z, Gagoshidze M, Nikolishvili M. Stability and convergence of the variable directions difference scheme for one nonlinear two-dimensional model. Int J Biomath, 2015, 8:1550057, 21 pages
[9] Kiguradze Z, Nikolishvili M, Tabatadze B. Numerical resolution of one system of nonlinear partial differential equations. Rep Enlarged Sess Semin I Vekua Appl Math, 2011, 25:76-79
[10] Douglas J. On the numerical integration of u_xx + u_yy=u_t by implicit methods. J Soc Industr Appl Math, 1955, 3:42-65
[11] Douglas J, Peaceman D W. Numerical solution of two-dimensional heat flow problems. Amer Inst Chem Engin J, 1955, 1:505-512
[12] Peaceman D W, Rachford H H. The numerical solution of parabolic and elliptic differential equations. J Soc Industr Appl Math, 1955, 3:28-41
[13] Janenko N N. The Method of Fractional Steps for Multi-dimensional Problems of Mathematical Physics (Russian). Moscow:Nauka, 1967
[14] Marchuk G I. The Splitting-up Methods (Russian). Moscow:Nauka, 1988
[15] Samarskii A A. The Theory of Difference Schemes (Russian). Moscow, 1977
[16] Abrashin V N. A variant of the method of variable directions for the solution of multi-dimensional problems in mathematical physics. I (Russian) Differ Uravn, 1990, 26:314-323; English translation:Differ Equ, 1990, 26:243-250

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

ECONOMICAL DIFFERENCE SCHEME FOR ONE MULTI-DIMENSIONAL NONLINEAR SYSTEM

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

Metrics

本文评价

推荐阅读 5

[1]	Nermina MUJAKOVIC, Nelida CRNJARIC-ZIC. GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY[J]. 数学物理学报(英文版), 2016, 36(6): 1541-1576.
[2]	袁益让, 程爱杰, 羊丹平, 李长峰. THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION[J]. 数学物理学报(英文版), 2015, 35(6): 1547-1565.
[3]	高理平. STABILITY AND SUPER CONVERGENCE ANALYSIS OF ADI-FDTD FOR THE 2D MAXWELL EQUATIONS IN A LOSSY MEDIUM[J]. 数学物理学报(英文版), 2012, 32(6): 2341-2368.
[4]	张法勇, 郭柏灵. ATTRACTORS FOR FULLY DISCRETE FINITE DIFFERENCE SCHEME OF DISSIPATIVE ZAKHAROV EQUATIONS[J]. 数学物理学报(英文版), 2012, 32(6): 2431-2452.
[5]	Richard Liska, Mikhail Shashkov, Burton Wendroff. THE EARLY INFLUENCE OF PETER LAX ON COMPUTATIONAL HYDRODYNAMICS AND AN APPLICATION OF LAX-FRIEDRICHS AND#br# LAX-WENDROFF ON TRIANGULAR GRIDS IN LAGRANGIAN COORDINATES[J]. 数学物理学报(英文版), 2011, 31(6): 2195-2202.
[6]	袁益让. THE UPWIND FINITE DIFFERENCE METHOD FOR MOVING BOUNDARY VALUE PROBLEM OF COUPLED SYSTEM[J]. 数学物理学报(英文版), 2011, 31(3): 857-881.
[7]	H. Kim, M. Laforest, D. Yoon. AN ADAPTIVE VERSION OF GLIMM'S SCHEME[J]. 数学物理学报(英文版), 2010, 30(2): 428-446.
[8]	Xinfeng Liu, Qing Nie. SPATIALLY-LOCALIZED SCAFFOLD PROTEINS MAY FACILITATE TO TRANSMIT LONG-RANGE SIGNALS[J]. 数学物理学报(英文版), 2009, 29(6): 1657-1669.
[9]	Carl L. Gardner, Steven J. Dwyer. NUMERICAL SIMULATION OF THE XZ TAURI SUPERSONIC ASTROPHYSICAL JET[J]. 数学物理学报(英文版), 2009, 29(6): 1677-1683.