ADI as a preconditioning for solving the convection-diffusion equation

Journal Article · Fri Jun 01 04:00:00 EDT 1984 · SIAM J. Sci. Stat. Comput.; (United States)

DOI:https://doi.org/10.1137/0905020· OSTI ID:5674000

Chin, R C.Y.; Manteuffel, T A; De Pillis, J

The authors examine a splitting of the operator obtained from a steady convection-diffusion equation with variable coefficients in which the convection term dominates. The operator is split into a dominant and subdominant parts consistent with the inherent directional property of the partial differential equation. The equations involving the dominant parts should be easily solved. The authors accelerate the convergence of this splitting or the outer iteration by a Chebyshev semi-iterative method. When the dominant part has constant coefficients, it can be easily solved using alternating direction implicit (ADI) methods. This is called the inner iteration. The optimal parameters for a stationary two-parameter ADI method are obtained when the eigenvalues become complex. This corresponds to either the horizontal or the vertical half-grid Reynolds number larger than unity. The Chebyshev semi-iterative method is used to accelerate the convergence of the inner ADI iteration. A two-fold increase in speed is obtained when the ADI iteration matrix has real eigenvalues, and the increase is less significant when the eigenvalues are complex. If either the horizontal or the vertical half-grid Reynolds number is equal to one, the spectral radius of the optimal ADI iterative matrix is zero. However, a high degree of nilpotency impairs rapid convergence. This problem is removed by introducing a more implicit iterative method called ADI/Gauss-Seidel (ADI/GS). ADI/GS resolves the nilpotency and, thus, converges more rapidly for half-grid Reynolds number near 1. Finally, these methods are compared with several well-known schemes on test problems. 23 references, 5 figures.

• Research Organization: Lawrence Livermore National Lab., CA
• DOE Contract Number: W-7405-ENG-48; W-7405-ENG-36
• OSTI ID: 5674000
• Journal Information: SIAM J. Sci. Stat. Comput.; (United States), Journal Name: SIAM J. Sci. Stat. Comput.; (United States) Vol. 5:2; ISSN SIJCD
• Country of Publication: United States
• Language: English

Related Subjects

640410 -- Fluid Physics-- General Fluid Dynamics
658000* -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
BOUNDARY-VALUE PROBLEMS
CONVECTION
DATA
DIFFUSION
ENERGY TRANSFER
HEAT TRANSFER
INFORMATION
ITERATIVE METHODS
MASS TRANSFER
MATRICES
NUMERICAL DATA
OPTIMIZATION
PERTURBATION THEORY
SINGULARITY
THEORETICAL DATA