An efficient and accurate operating splitting method for solving 3D heat conduction equations

Abstract

Abstract: The operator splitting local one-dimensional method is employed to resolve three-dimensional (3D) heat conduction equations. Two kinds of Padé approximations are used to calculate the temporal derivative, and two kinds of high-order difference formulas are used to calculate the spatial derivatives. Then, two kinds of efficient splitting local one-dimensional schemes are derived. The accuracy of them are the fourth-and sixth-order, respectively. The unconditional stability of both methods is proved by the rigorous matrix analysis theory. Numerical examples are resolved to validate the performances of the present method.

Key words: heat conduction equation, operator splitting, local one-dimensional scheme, high accuracy, unconditional stability

CLC Number:

O241.82

WU Li-li. An efficient and accurate operating splitting method for solving 3D heat conduction equations[J]. Journal of Lanzhou University of Technology, 2020, 46(5): 166-172.

References

[1] LIN Y,GAO X J,XIAO M Q.A high-order finite difference method for 1D nonhomogeneous heat equation[J].Numer Methods Partial Differential Eq,2009,25(2):327-346.
[2] 冯新龙,王焕焕,阿不都热西提.求解一维热传导方程数值解的高精度方法[J].新疆大学学报,2000,17(3):13-18.
[3] 田敏,羊丹平.热传导方程的一类新型重叠型并行区域分解有限差分算法[J].高等学校计算数学学报,2008,30(1):14-25.
[4] WANG T.Alternating direction finite volume element methods for 2D parabolic partial differential equations[J].Numer Methods Partial Differential Eq,2007,24(1):24-40.
[5] GANESAN S,TOBISKA L.Operator-splitting finite element algorithms for computations of high-dimensional parabolic problems[J].Appl Math Comput,2013,219(11):6182-6196.
[6] 欧阳华江.求解热传导方程的有限元算法[J].工程力学,1993,10(4):58-61.
[7] KOUATCHOU J.Comparison of time and spatial collocation methods for the heat equation[J].J Comput Appl Math,2003,150(1):129-141.
[8] BIALECKI B,DE FRUTOS J.ADI spectral collocation methods for parabolic problems[J].J Comput Phys,2010,229(13):5182-5193.
[9] PEACEMAN D W,RACHFORD H H.The numerical solution of parabolic and elliptic differential equation[J].J Soc Indust Appl Math,1955,3(1):28-41.
[10] DOUGLAS J,GUNN J E.A general formulation of alternating direction methods Part Ⅰ:parabolic and hyperbolic problems[J].Numer Math,1964,6(1):428-453.
[11] LI J,CHEN Y,LIU G.High order compact ADI methods for parabolic equations[J].Comput Math Appl,2006,52(2):1343-1356.
[12] QIN J.The new alternating direction implicit difference methods for solving three-dimensional parabolic equations[J].Appl Math Model,2010,34(4):890-897.
[13] DAI W Z,NASAR R.Compact finite difference scheme for solving parabolic differential equations[J].Numer Methods Partial Differential Eq,2002,18(2):129-141.
[14] ZHOU H,WU Y J,TIAN W.Extrapolation algorithm of compact ADI approximation for two-dimensional parabolic equations[J].Appl Math Comput,2012,219(6):2875-2884.
[15] GAO G H,SUN Z Z.Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations[J].J Scientific Comput,2016,66(3):1281-1312.
[16] 葛永斌,田振夫,吴文权.高维热传导方程的高精度交替方向隐式方法[J].上海理工大学学报,2007,29(1):55-58.
[17] WANG Y M.Error and extrapolation of a compact LOD method for parabolic differential equations[J].J Comput Appl Math,2011,235(5):1367-1382.
[18] WANG T,WANG Y M.A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations[J].Appl Math Comput,2014,237:512-530.
[19] DOUGLAS J,KIMY S.Improved accuracy for locally one-dimensional methods for parabolic equations[J].Math Models Methods Appl Sci,2001,11(2):1563-1579.
[20] KARAA S.An accurate LOD scheme for two-dimensional parabolic problems[J].Appl Math Comput,2005,170(2):886-894.
[21] ZHANG S H,LIANG D.Locally one-dimensional-alternating segment explicit-implicit and locally one-dimensional-alternating segment Crank-Nicolson methods for two-dimension parabolic equations[J].Intern J Computer Math,2015,92(3):513-531.
[22] HIRSH R S.High order accurate difference solutions of fluid mechanics problem by a compact differencing technique[J].J Comput Phys,1975,19(1):90-109.
[23] CUYT A.Padé Approximants for operators:theory and applications[M].Berlin:Springer,1984.