By Rob F. Remis, Neil V. Budko (auth.), Barry Koren, Kees Vuik (eds.)
The goal of the current e-book is to teach, in a large and but deep manner, the cutting-edge in computational technological know-how and engineering. Examples of themes addressed are: quick and exact numerical algorithms, model-order aid, grid computing, immersed-boundary tools, and particular computational tools for simulating a wide selection of demanding difficulties, difficulties equivalent to: fluid-structure interplay, turbulent flames, bone-fracture therapeutic, micro-electro-mechanical structures, failure of composite fabrics, typhoon surges, particulate flows, and so forth. the most profit provided to readers of the booklet is a well-balanced, updated review over the sector of computational technological know-how and engineering, via in-depth articles through experts from the separate disciplines.
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Additional info for Advanced Computational Methods in Science and Engineering
O OSTERLEE , C. V UIK , A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27: 1471-1492, 2006. Shifted-Laplacian Preconditioners for Heterogeneous Helmholtz Problems 45 19. A. E RLANGGA , Advances in Iterative Methods and Preconditioners for the Helmholtz Equation Archives Comput. , 15: 37-66, 2008. 20. M. B. VAN G IJZEN , Y. A. E RLANGGA , C. V UIK , Spectral Analysis of the Discrete Helmholtz Operator Preconditioned with a Shifted Laplacian SIAM J.
5), see also ; (a) k = 40, h = 1/64 and (b) k = 100, h = 1/160. This is beneficial for iterative solution methods. From the spectra in Figure 6 it is expected that the Bi-CGSTAB (and GMRES) convergence in the case of damping in the original equation will be considerably faster than in the undamped case. As the circles have moved away from the origin it is possible to apply the classical theory of the GMRES convergence [52, 53], for example. Returning to the undamped case, α = 0, we concentrate on the choice of discretization, and fix k = 100 (k2 = 104 ), h = 1/160.
We perform Fourier analysis here to visualize the effect of the choice of the parameters, (β1 , β2 ), as well as the choice of discretization on the clustering of the eigenvalues of the preconditioned system. This analysis gives only a first indication of what we can expect from the solver. For both Ah and Mh , we use either the secondorder discretizations or the fourth-order, HO stencils. Initially, we do not include damping in Ah in the analysis (we take α = 0 in (1)). First, we visualize the effect of the choice of (β1 , β2 ) in the preconditioner on the clustering of the eigenvalues of the preconditioned system.