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

Abstract

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.

High-order compact exponential finite difference methods for convection–diffusion type problems

“…The convergence and performance of iterative methods with HOC polynomial schemes have been studied in [18,38,39]. The convergence and performance of iterative methods with EHOC schemes have not yet to be investigated.…”

“…Straightforwardly calculating f 1x , f 1xx , f 1y and f 2yy and substituting into the right-hand sides of (39) and (40), and then combining (39) and (40) and rearranging, we obtain an O(h 4 + k 4 ) compact exponential FD approximation to (31) at a mesh point (x i , y j ) as…”

Accuracy, robustness and efficiency comparison in iterative computation of convection diffusion equation with boundary layers

“…In the present work, we use a standard multigrid approach with an alternating line Gauss-Seidel relaxation [17]. Smoothing analyses in [15] show that the alternating line Gauss-Seidel relaxation is a robust smoother for the convection diffusion equation discretized by FCS. One presmoothing sweep and one postsmoothing sweep are performed on each level in a V-cycle algorithm.…”

“…(1) with boundary layers, for the sake of saving space, we refer readers to the original article for the details of the formula and the derivation procedure [6]. This discretization scheme has been shown to be computationally efficient and numerically stable with respect to the application of iterative techniques, in addition to producing high-accuracy approximate solutions for smooth functions [7,2,15]. The unconditional stability of FCS makes it very attractive in use with the multigrid method [2].…”

On a High-Order Compact Scheme and Its Utilization in Parallel Solution of a Time-Dependent System on a Distributed Memory Processor

“…In summary, a compact differencing scheme requires more work per grid point, but the result is higher approximation accuracy, a few grid points to compute with, and less computer memory requirements to store the computed result. As a result, compact approximation schemes are more efficient than both non-compact methods of the same order and also than low-order solution methods in general [18,19,20]. As stated by Orszak [21], at the cost of slight computational complexity, fourth-order compact schemes can achieve results in the 5% accuracy range with approximately half the spatial resolution in each space direction when compared with the second-order schemes.…”