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In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.
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篇名 Two-Level Block Decompositions for Solving Helmholtz Equation via Chebyshev Pseudo Spectral Method
来源期刊 现代物理(英文) 学科 数学
关键词 Helmholtz Equation CHEBYSHEV Pseudo-Spectral Method CHEBYSHEV Differentiation MATRIX Coarse-Grain PARALLELISM REFLEXIVE MATRIX
年,卷(期) 2018,(9) 所属期刊栏目
研究方向 页码范围 1713-1723
页数 11页 分类号 O1
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Helmholtz
Equation
CHEBYSHEV
Pseudo-Spectral
Method
CHEBYSHEV
Differentiation
MATRIX
Coarse-Grain
PARALLELISM
REFLEXIVE
MATRIX
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研究去脉
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期刊影响力
现代物理(英文)
月刊
2153-1196
武汉市江夏区汤逊湖北路38号光谷总部空间
出版文献量(篇)
1826
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0
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0
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