Applied and Computational Mathematics
Volume 2, Issue 3, June 2013, Pages: 64-77
Received: May 13, 2013;
Published: Jun. 30, 2013
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Rabab Ahmed Shanab, Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt
Laila Fouad Seddek, Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt
Salwa Amin Mohamed, Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Egypt
In this paper, we extend the work of Kalita et al.  to solve the steady 3D convection-diffusion equation with variable coefficients on non-uniform grid. The approach is based on the use of Taylor series expansion, up to the fourth order terms, to approximate the derivatives appearing in the 3D convection diffusion equation. Then the original convection-diffusion equation is used again to replace the resulting higher order derivative terms. This leads to a higher order scheme on a compact stencil (HOC) of nineteen points. Effectiveness of this method is seen from the fact that it can handle the singularity perturbed problems by employing a flexible discretized grid that can be adapted to the singularity in the domain. Four difficult test cases are chosen to demonstrate the accuracy of the present scheme. Numerical results show that the fourth order accuracy is achieved even though the Reynolds number (Re) is high.
Rabab Ahmed Shanab,
Laila Fouad Seddek,
Salwa Amin Mohamed,
Non-uniform HOC Scheme for the 3D Convection–Diffusion Equation, Applied and Computational Mathematics.
Vol. 2, No. 3,
2013, pp. 64-77.
S. KARAA, J. Zhang, Convergence and Performance of Iterative Methods for Solving Variable Coefficient Convection-Diffusion Equation with a Fourth-Order Compact Difference Scheme , Computers and Mathematics with Applications 44 (2002) 457-479.
J. Zhang , Numerical Simulation of 2D Square Driven Cavity Using Fourth-Order Compact Finite Difference Schemes, Computers and Mathematics with Applications 45 (2003) 43-52.
Y. Wang, J. Zhang, Integrated fast and high accuracy computation of convection diffusion equations using multiscale multigrid method, Numerical Methods for Partial Differential Equations, 27(2):399-414, 2011
L. Ge, J. Zhang, Symbolic computation of high order compact difference for three dimensional linear elliptic partial differential equations with variable schemes coefficients, Journal of Computational and Applied Mathematics 143 (2002) 9–27
M. gupta, J. zhang, High accuracy multigrid solution of the 3D Convection Diffusion equation, Applied Mathematics and Computation 113 (2000) 249–274
Y. Wang, J. Zhang, Fast and robust sixth-order multigrid computation for the three-dimensional Convection-Diffusion equation, Journal of Computational and Applied Mathematics 234, pp.3496-3506, 2010.
J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional Convection–Diffusion equation, Communications in Numerical Methods in Engineering, Vol. 14, 263--280 (1998).
Y. Ge, Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation, Journal of Computational Physics 229 (2010) 6381–6391
Y. Ma, Y. Ge, A high order finite difference method with Richardson extrapolation for 3D Convection Diffusion equation, Applied Mathematics and Computation 215 (2010) 3408–3417
L. Ge, J, Zhang, High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Non-uniform Grids, Journal of Computational Physics171,560–578 (2001)
J. Kalita, A. Dass, D. Dalal, A transformation-free HOC scheme for steady Convection–Diffusion on non-uniform grids, Int. J. Numer. Meth. Fluids 44 (2004) 33–53
Y. Ge, F. Cao, Multigrid method based on the transformation-free HOC scheme on non-uniform grids for 2D Convection Diffusion problems, Journal of Computational Physics 230 (2011) 4051–4070
Y. Ge, F. Cao, J. Zhang, A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on non-uniform grids, Journal of Computational Physics (2012)
A. Brandt, Multilevel Adaptive Solutions to Boundary Value Problems, Mathematics of Computation, Vol.31, pp. 333–390, 1977.
W. Hachbusch, V. Trottenberg, Multigrid Methods Lecture Notes in Mathematics, Vol. 960, Springer: Berlin, 1982.
W. L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987.
P. Wesseling, An Introduction to Multigrid Methods, Weily, England, 1992.
K. Stuben, A Review of Algebraic Multigrid, Journal of Computational and Applied Mathematics, Vol.128, pp.281–309, 2001.
C. Tien Wu and H. C. Elman, Analysis and Comparison of Geometric and Algebraic Multigrid for Convection‐Diffusion Equations, SIAM J. Sci. Comput., Vol.28, pp.2208–2228, 2006.
V. Gravemeier, M. W. Gee and W.A. Wall, "An algebraic variational multiscale–multigrid method based on plain aggregation for convection–diffusion problems", Comput. Methods Appl. Mech. Engrg. Vol.198, pp.3821–3835, 2009.
W. Briggs, V. Henson, and S. McCormick, A Multigrid Tutorial, SIAM, Philadelphia, 2000. [Chapter 8: Algebraic Multigrid (AMG)]
J. Ruge and K. Stuben, Algebraic multigrid (AMG), in Multigrid Methods, Frontiers in Applied Mathematics, S. F. McCormick, ed., SIAM, Philadelphia, 1987, pp. 73–130.
U. Trottenberg, C. Oosterlee, and A. Schuller, Multigrid, Academic Press, London, 2001. [appendix A: Algebraic Multigrid (AMG) by K. Stuben]