A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations

作者：Chen Feng^*

来源：Journal of Scientific Computing, 2015, 62(2): 575-600.

DOI：10.1007/s10915-014-9868-3

• Spectral collocation
• Spectral-Galerkin
• System
• Coupled
• GPU
• CUDA
• Cahn-Hilliard
• FitzHugh-Nagumo

Spectral methods are useful for applications that benefit from high-order precisions. However, if the same number of degrees of freedom is used, the computational cost of a spectral method is considerably higher than that of a general finite difference or finite element method. After the investigation in Chen et al. (J Comput Phys 250:555-564, 2013), we provide for the first time a framework of graphics processing units (GPU)-accelerated spectral methods for systems of coupled elliptic equations. The involved dense matrix computations, as the main obstacle for fast spectral methods on a traditional CPU, turns out to be an opportunity for high speedups on a many-core GPU. We obtain an order-of-magnitude speedup for solving 2-D and 3-D systems using a Kepler 20 GPU over a high-end multi-core processor, with two popular spectral methods, namely, the spectral collocation method and the spectral-Galerkin method. The new framework is applicable to systems of coupled second-order equations with general boundary conditions, where is an integer of moderate size. The ultimate goal is to apply the developed solver to complex and nonlinear time-dependent problems. As two interesting examples, a 2-D FitzHugh-Nagumo equation is solved with the spectral collocation method and a 3-D Cahn-Hilliard equation is solved with the spectral-Galerkin method. We thus demonstrate a practical solution for demanding problems that utilize high-order spatial resolution and longer run-times.

• 出版日期2015-2