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An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains
Computer Physics Communications ( IF 6.3 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.cpc.2020.107695
Max Carlson , Xiaoning Zheng , Hari Sundar , George Em Karniadakis , Robert M. Kirby

Abstract We present a spectral element algorithm and open-source code for computing the fractional Laplacian defined by the eigenfunction expansion on finite 2D/3D complex domains with both homogeneous and nonhomogeneous boundaries. We demonstrate the scalability of the spectral element algorithm on large clusters by constructing the fractional Laplacian based on computed eigenvalues and eigenfunctions using up to thousands of CPUs. To demonstrate the accuracy of this eigen-based approach for computing the factional Laplacian, we approximate the solutions of the fractional diffusion equation using the computed eigenvalues and eigenfunctions on a 2D quadrilateral, and on a 3D cubic and cylindrical domain, and compare the results with the contrived solutions to demonstrate fast convergence. Subsequently, we present simulation results for a fractional diffusion equation on a hand-shaped domain discretized with 3D hexahedra, as well as on a domain constructed from the Hanford site geometry corresponding to nonzero Dirichlet boundary conditions. Finally, we apply the algorithm to solve the surface quasi-geostrophic (SQG) equation on a 2D square with periodic boundaries. Simulation results demonstrate the accuracy, efficiency, and geometric flexibility of our algorithm and that our algorithm can capture the subtle dynamics of anomalous diffusion modeled by the fractional Laplacian on complex geometry domains. The included open-source code is the first of its kind. Program summary Program title: Nektarpp_EigenMM CPC Library link to program files: http://dx.doi.org/10.17632/whtc75rj55.1 Developer’s repository link: https://github.com/paralab/Nektarpp_EigenMM Licensing provisions: MIT License Programming language: C/C++, MPI Nature of problem: An open-source parallel code for computing the spectral fractional Laplacian on 3D complex geometry domains. Solution method: A distributed, sparse, iterative algorithm is developed to solve an associated integer-order Laplace eigenvalue problem for use in computing approximate solutions to the fractional diffusion equation. Additional comments including restrictions and unusual features: The code is implemented on CPUs with super-linear parallel efficiency at extreme scale.

中文翻译:

用于计算 3D 复杂几何域上的光谱分数拉普拉斯算子的开源并行代码

摘要 我们提出了一种谱元算法和开源代码,用于在具有齐次和非齐次边界的有限 2D/3D 复杂域上计算由特征函数展开定义的分数拉普拉斯算子。我们通过使用多达数千个 CPU 构建基于计算特征值和特征函数的分数拉普拉斯算子,证明了光谱元素算法在大型集群上的可扩展性。为了证明这种基于特征的方法计算派系拉普拉斯算子的准确性,我们在 2D 四边形、3D 立方和圆柱域上使用计算的特征值和特征函数来近似分数扩散方程的解,并将结果与人为的解决方案来证明快速收敛。随后,我们在用 3D 六面体离散的手形域上以及在由对应于非零 Dirichlet 边界条件的 Hanford 站点几何构造的域上呈现分数扩散方程的模拟结果。最后,我们应用该算法在具有周期性边界的二维正方形上求解表面准地转 (SQG) 方程。仿真结果证明了我们算法的准确性、效率和几何灵活性,并且我们的算法可以捕获由分数拉普拉斯算子在复杂几何域上建模的异常扩散的微妙动态。包含的开源代码是同类中的第一个。程序摘要 程序名称:Nektarpp_EigenMM CPC 库程序文件链接:http://dx.doi.org/10.17632/whtc75rj55.1 开发者存储库链接:https://github。com/paralab/Nektarpp_EigenMM 许可条款:MIT 许可 编程语言:C/C++,MPI 问题性质:用于计算 3D 复杂几何域上的光谱分数拉普拉斯算子的开源并行代码。求解方法:开发了一种分布式、稀疏、迭代算法来求解相关的整数阶拉普拉斯特征值问题,用于计算分数扩散方程的近似解。包括限制和不寻常功能在内的其他评论:该代码在具有超线性并行效率的 CPU 上以极端规模实现。迭代算法被开发来解决相关的整数阶拉普拉斯特征值问题,用于计算分数扩散方程的近似解。包括限制和不寻常功能在内的其他评论:该代码在具有超线性并行效率的 CPU 上以极端规模实现。迭代算法被开发来解决相关的整数阶拉普拉斯特征值问题,用于计算分数扩散方程的近似解。包括限制和不寻常功能在内的其他评论:该代码在具有超线性并行效率的 CPU 上以极端规模实现。
更新日期:2021-04-01
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