Archive
Special Issues

Volume 7, Issue 1, February 2018, Page: 1-10
A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems
Yongbin Ge, Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
Fei Zhao, Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
Jianying Wei, Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China
Received: Dec. 3, 2017;       Accepted: Dec. 13, 2017;       Published: Jan. 12, 2018
Abstract
In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].
Keywords
3D Unsteady Convection Diffusion Equation, Rational, High Order Compact Scheme, ADI Method, Stability
Yongbin Ge, Fei Zhao, Jianying Wei, A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems, Applied and Computational Mathematics. Vol. 7, No. 1, 2018, pp. 1-10. doi: 10.11648/j.acm.20180701.11
Reference
[1]
J. Bear, Dynamics of Fluids of Porous Media, American Elsevier Publishing Company, New York, 1972.
[2]
S. V. Patankar, Numerical Heat Transfer and Fluid Flows, McGraw-Hill, New York, 1980.
[3]
Z. Zlatev, R. Berkowicz, L. P. Prahm, Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants, J. Comput. Phys. 55 (1984) 278-301.
[4]
R. S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975) 90-109.
[5]
B. J. Noye, H. H. Tan, Finite difference methods for solving the two-dimensional advection diffusion equation, Int. J. Numer. Meth. Fluids 9 (1988) 75-89.
[6]
A. Rigal, High order difference schemes for one-dimensional convection-diffusion problems, J. Comput. Phys. 114 (1994) 59-76.
[7]
K. W. Morton, Numerical Solution of Convection-diffusion Problems, Chapman & Hall, London, 1996.
[8]
B. J. Noye, H. H. Tan, A third-order semi-implicit finite difference methods for solving the one-dimensional unsteady convection diffusion equation, Int. J. Numer. Meth. Eng. 26 (1998) 1615-1629.
[9]
W. F. Spotz, G. F. Carey, Extension of high-order compact schemes to time-dependent problems, Numer. Meth. Partial Differential Eq.17 (2001) 657-672.
[10]
J. C. Kalita, D. C. Dalal, A. K. Dass, A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients, Int. J. Numer. Meth. Fluids 38 (2002) 1111-1131.
[11]
Z. F. Tian, A rational high-order compact ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun. 182 (2011) 649-662.
[12]
Z. F. Tian, P. X. Yu, A high-order exponential scheme for solving 1D unsteady convection- diffusion equations, J. Comput. Appl. Math. 235 (2011) 2477-2491.
[13]
J. C. Kalita, S. Sen, An improved (9,5) higher order compact schemes for the transient two-dimensional convection-diffusion equation, Int. J. Numer. Meth. Fluids 51 (2006) 703 -717.
[14]
J. C. Kalita, S. Sen, The (9,5) HOC formulation for the transient Navier-Stokes equations in primitive variable, Int. J. Numer. Meth. Fluids 55 (2007) 387-406.
[15]
D. W. Peaceman, H. H. Rachford Jr., The numerical of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1959) 28-41.
[16]
J. Douglas Jr., J. E. Gunn, A general formulation of alternating direction methods. I. Parabolic and hyperbolic problems, Numer. Math. 6 (1964) 428-453.
[17]
M. Dehghan, A. Mohebbi, High-order compact boundary value method for the solution of unsteady convection-diffusion problems, Math. Comput. Simul.79 (2008) 683-699.
[18]
J. Qin, The new alternating direction implicit difference methods for solving three-dimension parabolic equations, Appl. Math. Modelling, 34 (2010) 890-897.
[19]
A. Mohebbi, M. Dehghan, High-order compact solution of the one-dimensional heat and advection-diffusion equations, Appl. Math. Modelling, 34 (2010) 3071-3084.
[20]
S. Karaa, J. Zhang, High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys. 198 (2004) 1-9.
[21]
D. You, A high-order Padé ADI method for unsteady convection-diffusion equations, J. Comput. Phys. 214 (2006) 1-11.
[22]
Z. F. Tian, Y. B. Ge, A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math. 198 (2007) 268-286.
[23]
H. W. Sun, L. Z. Li, A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun. 185 (2014) 790-797.
[24]
M. Dehghan, Numerical solution of the three-dimensional advection-diffusion equation, Appl. Math. Comput. 150 (2004) 5-19.
[25]
S. D. Wang, Y. M. Shen, Three high-order splitting schemes for 3D transport equation, Appl. Math. Mech. 26 (2005) 1007-1016.
[26]
S. Karaa, A high-order compact ADI method for solving three-dimensional unsteady convection-diffusion problems, Numer. Meth. Partial Differential Eq. 22 (2006) 983-993.
[27]
F. Cao, Y. Ge. A high-order compact ADI scheme for the 3D unsteady convection-diffusion equation. 2011 International Conference on Computational and Information Sciences. 2011, PP1087-1089.
[28]
Y. Ge, Z. F. Tian, J. Zhang, An exponential high-order compact ADI method for 3D unsteady convection-diffusion problems, Numer. Meth. Partial Differential Eq. 29 (2013) 186-205.
[29]
J. Qin, T. Wang, A compact locally one-dimensional finite difference method for nonhomogeneous parabolic differential equations, Int. J. Numer. Meth. Biomed. Engng. 7 (2011) 128-142.
[30]
M. Dehghan, Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition, Math. Comput. Simulat. 49 (1999) 331-349.
[31]
S. Karaa, An accurate LOD scheme for two-dimensional parabolic problems, Appl Math Comput, 170 (2005) 886-894.
[32]
W. Zhang, L. Tong, E. T. Chung, A new high accuracy locally one-dimensional scheme for the wave equation, J. Comput. Appl. Math. 236 (2011) 1343-1353.
[33]
E. G. D’yakonov, Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed derivatives, (Russian) Zh. Vychisl. Mat. I Mat. Fiz 4 (1964) 935-941.
[34]
Y. Sun, Z. J. Wang, Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grids, Int. J. Numer. Meth. Fluids 45 (2004) 819-838.
[35]
A. Kaya, Finite difference approximations of multidimensional unsteady convection-diffusion -reaction equations. J. Comput. Phys. 285 (2015) 331-349.
[36]
Jiang T, Zhang Y T. Krylov single-step implicit integration factor WENO methods for advection-diffusion-reaction equations. J. Comput. Phys. 311 (2016) 22-44.