A Hierarchical 3-D Direct Helmholtz Solver by Domain Decomposition and Modified Fourier Method

Abstract

The paper contains a non-iterative solver for the Helmholtz and the modified Helmholtz equations in a hexahedron. The solver is based on domain decomposition. The solution domain is divided into mostly parallelepiped subdomains. In each subdomain a particular solution of the non-homogeneous Helmholtz equation is first computed by a fast spectral 3-D method which was developed in our earlier papers (see, for example, SIAM J. Sci. Comput. 20 (1999Comput. 20 ( ), 2237Comput. 20 ( -2260. This method is based on the application of the discrete Fourier transform accompanied by a subtraction technique. For high accuracy the subdomain boundary conditions must be compatible with the specified inhomogeneous right hand side at the edges of all the interfaces. In the following steps the partial solutions are hierarchically matched. At each step pairs of adjacent subdomains are merged into larger units. The paper describes in detail the matching algorithm for two boxes which is a basis of the domain decomposition scheme. The hierarchical approach is convenient for parallelization and can minimize the global communication. The algorithm requires O(N 3 log N ) operations, where N is the number of grid points in each direction.

Isolation, Synthesis, and Biological Activity of Aphrocallistin, an Adenine-Substituted Bromotyramine Metabolite from the Hexactinellida Sponge Aphrocallistes beatrix

“…Desulfuration of 12 in the presence of Raney Ni 8,10 led to intermediate 13. Reaction of the purin-6-one 13 with phosphorus pentasulfide, 10 alkylation on the sulfur residue, 10 and subsequent chloride displacement 11 formed 14, a key intermediate for the final step of the synthesis of aphrocallistin (1).…”

Parallel calculations in the Darwin PIC-model

“…В дополнение к этому отдельной проблемой становится решение (особенно при его итерационном характере [Бородачев, Коломиец, Литвинюк, 2006]) полевых уравнений, требующее теперь либо интенсивных межпроцессорных обменов сеточными значениями, либо организации внутреннего итерационного процесса с обменом граничными сеточными значениями соседних подобластей [Rice, Vavalis, Yang, 1997;Braverman, Israeli, Averbuch, 2005]. Помимо значительного увеличения и без того немалых накладных коммуникационных расходов, последнее Л. В. Бородачев, Д. О. Коломиец актуализирует вопросы сходимости глобального сеточного решения.…”

An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

“…The use of such solvers in canonical geometries remains at the heart of many computing codes in physics. Examples are among others fluid dymanics (compressible or incompressible Navier-Stokes equations), [12][13][14], the Helmholtz equation [15], computations in astrophysics, [16] or in geophysics, [17]. The scheme referred as hermitian box-scheme was first introduced in [18].…”