A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

International Journal for Numerical Methods in Fluids - Tập 70 Số 6 - Trang 703-712 - 2012
Yanbao Ma1,2, Chien‐Pin Sun3, David A. Haake4,5, Bernard M. Churchill4, Chih‐Ming Ho3
1School of Engineering, University of California at Merced, Merced, CA 95343, USA
2Yanbao Ma, School of Engineering, University of California at Merced, Merced, CA 95343, USA.
3Department of Mechanical and Aerospace Engineering, University of California at Los Angeles, Los Angeles, CA 90095, USA
4Schools of Medicine, University of California at Los Angeles, Los Angeles, CA, 90095 USA
5VA Greater Los Angeles Healthcare System, Los Angeles, CA 90073, USA

Tóm tắt

SUMMARYA high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.

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International Journal for Numerical Methods in Fluids

Tập 70 Số 6

703-712

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