高效求解声硬散射中出现的时间域边界积分方程
Efficient Solution of Time-Domain Boundary Integral Equations Arising in Sound-Hard Scattering
AI总结：
本文提出一种高效的时空伽辽金方法，用于求解具有Neumann边界条件的三维波动方程，采用高光滑性、紧支撑基函数和分段多项式基函数，并探讨了系统矩阵的结构及其高效并行组装，提出多种预处理策略以解决块Hessenberg矩阵系统，并展示了并行实现的数值结果。
摘要：
我们考虑通过时间域边界积分方程高效数值求解具有Neumann边界条件的三维波动方程。采用具有$C^\infty$-smooth、紧支撑基函数在时间域，分段多项式基函数在空间域的时空伽辽金方法。讨论了系统矩阵的结构及其高效并行组装。针对出现的块Hessenberg矩阵系统，提出了不同的预处理策略并进行了数值研究。此外，还展示了一个使用OpenMP和MPI在共享和分布式内存中并行化的C++实现，该实现属于边界元库BEM4I。提供了包括收敛性和可扩展性测试在内的数值实验结果，直至千核集群。所提出的实现显示出良好的系统矩阵组装并行可扩展性。此外，所提出的代数预处理器与FGMRES求解器结合，显著减少了计算时间。
英文摘要：
We consider the efficient numerical solution of the three-dimensional wave equation with Neumann boundary conditions via time-domain boundary integral equations. A space-time Galerkin method with $C^\infty$-smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time.
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