Abstract
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 \(L\) coupled second-order equations with general boundary conditions, where \(L\) 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.
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Acknowledgments
The author first would like to thank the referees for helpful suggestions and comments. The author would also like to thank Jie Shen at Purdue University, Jan Hesthaven at the EPFL, Kevin Wang at Virginia Tech, and Michael Frank at Brown University for their help and insightful suggestions. The author acknowledges partial support by OSD/AFOSR FA9550-09-1-0613 and AFOSR FA9550-12-1-0463. The computing facility is provided by Division of Applied Mathematics and the Center for Computation and Visualization (CCV) at Brown University.
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Appendix: Sample CUDA Codes
Appendix: Sample CUDA Codes
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Chen, F. A New Framework of GPU-Accelerated Spectral Solvers: Collocation and Glerkin Methods for Systems of Coupled Elliptic Equations. J Sci Comput 62, 575–600 (2015). https://doi.org/10.1007/s10915-014-9868-3
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DOI: https://doi.org/10.1007/s10915-014-9868-3