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Chen, Feng

A new framework of GPU-accelerated spectral solvers: collocation and Galerkin methods for systems of coupled elliptic equations. (A new framework of GPU-accelerated spectral solvers: collocation and Glerkin methods for systems of coupled elliptic equations.) (English) Zbl 1320.65183

J. Sci. Comput. 62, No. 2, 575-600 (2015).
Summary: 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 [F. Chen and J. Shen, “A GPU parallelized spectral method for elliptic equations in rectangular domains”, 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.

Cited in 3 Documents

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35Q35 PDEs in connection with fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Keywords:

spectral collocation; spectral-Galerkin; system; coupled; GPU; CUDA; Cahn-Hilliard; Fitzhugh-Nagumo; numerical example

Software:

CUDA
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Full Text: DOI

References:

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