Analysis and parallel implementation of a forced N-body problem

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• 作者：C.E. Torres^c ; ^d ; H. Parishani^b ; O. Ayala^e ; L.F. Rossi^a ; ^{rossi@math.udel.edu} ; L.-P. Wang^b
• 关键词：GMRes ; Preconditioner ; Domain decomposition ; Hydrodynamic interaction ; Cloud droplets ; Direct numerical simulation ; MPI implementation ; Contour integral
• 刊名：Journal of Computational Physics
• 出版年：2013
• 出版时间：15 July, 2013
• 年：2013
• 卷：245
• 期：Complete
• 页码：235-258
• 全文大小：2915 K

文摘

The understanding of particle dynamics in N-body problems is of importance to many applications in astrophysics, molecular dynamics and cloud/plasma physics where the theoretical representation results in a coupled system of equations for a large number of entities. This paper concerns algorithms for solving a specific N-body problem, namely, a system of disturbance velocities for hydrodynamically interacting particles in a particle-laden turbulent flow. The system is derived from the improved superposition method of . Targeting for scalable computations on petascale computers, we have carried out a thorough study of a parallel implementation of GMRes with different features, such as preconditioners, matrix-free and parallel sparse representation of the matrix through 1D and 2D spatial domain decompositions. Gauss-Seidel method is also studied as a reference iterative algorithm. The range of conditions for efficiency and failure of each method is discussed in detail.

Through perturbation analysis, we have conducted a series of experiments to understand the effect of particle sizes, interaction symmetry, inter-particle distances and interaction truncation on the eigenvalues and normality of the linear system. For situations where the system is ill-conditioned, we introduce a restricted Schwarz type preconditioner. We verified the parallel efficiency of the preconditioner using 1D domain decomposition on a parallel machine. A benchmark problem of particle laden turbulence at resolution with particles is studied to understand the scalability of the proposed methods on parallel machines. We have developed a stable and highly scalable parallel solver with an affordable computational cost even for ill-conditioned systems through preconditioning. On 64 cores, using GMRes in 2D domain decomposition, we achieved a speed-up of x (relative to 1D domain decomposition on the same number of processors). Our complexity analysis showed that for large N-body problems, the proposed GMRes scheme scales well for moderate to large number of processors in current tera to petascale computers.