Convergence of iterative methods for a fourth-order discretization scheme

Numer. Methods Partial Differential Eq.

2000

We present a new strategy to accelerate the convergence rate of a high-accuracy multigrid method for the numerical solution of the convection-diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss-Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourthorder compact discretization are investigated numerically.

Section: B Diagonal Dominancementioning

confidence: 99%

A note on an accelerated high-accuracy multigrid solution of the convection-diffusion equation with high Reynolds number

Numer. Methods Partial Differential Eq.

2000

“…All these schemes may look slightly different, but all reported numerical results are similar. The convergence of these schemes with the basic iterative methods has been verified numerically, but rigorous justification of convergence and systematic study on the performance of these highaccuracy schemes with fast iterative methods are in their very early stages [3,9,14].…”

Section: Two-dimensional Problemmentioning

confidence: 99%

“…It was proved in [9] that FOCS with point Jacobi and Gauss-Seidel methods converges for |δ| ≤ 1 and |γ| ≤ 1 and the spectral radius of the line Jacobi iteration matrix with FOCS is bounded by (17 + 20 √ 2)/73 when |δ| = |γ| = √ 2. In the limit of pure diffusion (δ, γ → 0), (11) (and all other FOCS for the 2D convectiondiffusion equation) reduces to the Mehrstellen operator.…”

Section: Fig 2 Contours Of the Error Reduction Factor Of The Point mentioning

confidence: 99%

“…(9). α is the so-called flow angle, which measures the relative angle formed between the grid line (x-axis) and the flow characteristics.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Basic iterative methods with this scheme have been shown (numerically) to converge for large values of the convection coefficients. In Zhang [9], we analyzed the convergence of some basic iterative methods with this scheme with small Reynolds numbers. A systematic development of the high-order compact schemes was investigated by Spotz [10].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On convergence and performance of iterative methods with fourth-order compact schemes

Numer. Methods Partial Differential Eq.

1998

We study the convergence and performance of iterative methods with the fourth-order compact discretization schemes for the one-and two-dimensional convection-diffusion equations. For the one-dimensional problem, we investigate the symmetrizability of the coefficient matrix and derive an analytical formula for the spectral radius of the point Jacobi iteration matrix. For the two-dimensional problem, we conduct Fourier analysis to determine the error reduction factors of several basic iterative methods and comment on their potential use as the smoothers for the multilevel methods. Finally, we perform numerical experiments to verify our Fourier analysis results.

An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation

Commun. Numer. Meth. Engng.

1998

SUMMARYWe present an explicit fourth-order compact ®nite dierence scheme for approximating the threedimensional convection±diusion equation with variable coecients. This 19-point formula is de®ned on a uniform cubic grid. We compare the advantages and implementation costs of the new scheme with the standard 7-point scheme in the context of basic iterative methods. Numerical examples are used to verify the fourth-order convergence rate of the scheme and to show that the Gauss±Seidel iterative method converges for large values of the convection coecients. Some algebraic properties of the coecient matrices arising from dierent discretization schemes are compared. We also comment on the potential use of the fourthorder compact scheme with multilevel iterative methods. #