GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

Vargas, Anielle de; Chan, Jesse; Hagstrom, Thomas; Warburton, Tim

doi:10.1007/978-3-319-65870-4_25

Cited by 3 publications

(13 citation statements)

References 14 publications

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“…Noticeably a single GPU kernel offers a better time to solution with advantage of reliving the need to store the interpolate on account of less data movement. These results are consistent with what has been observed in [14].…”

Section: Acceleration On Graphics Processing Unitssupporting

confidence: 94%

“…A full description of the interpolation procedure can be found in [7,13,15]. The coefficients of the polynomial are used to carry out the time stepping algorithm.…”

Section: Description Of the Methodsmentioning

confidence: 99%

“…Given the tensor structure of the polynomial, the interpolant enables a dimension by dimension reconstruction. Figure 7 illustrates the staggering of the nodes as well as the reconstruction procedure; we refer the reader to [13] for further details the higher dimensional reconstruction procedure. Fig.…”

Section: Extending the Methods To Higher Dimensionsmentioning

confidence: 99%

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Leapfrog Time-Stepping for Hermite Methods

et al. 2019

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We introduce Hermite-leapfrog methods for first order wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors [7]. The new schemes stagger field variables in both time and space and are high-order accurate. We provide a detailed description of the method and demonstrate that the method conserves variable quantities in one-space dimension. Higher dimensional versions of the method are constructed via a tensor product construction. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented.Keywords High order · Hermite · Leapfrog IntroductionSimulations of wave propagation play a crucial role in science and engineering. In applications to geophysics, they are the engine of many seismic imaging algorithms. For electrical engineers, they can be a useful tool for the design of radars and antennas. In these applications achieving high fidelity simulations is challenging due to the inherent issues in modeling highly oscillatory waves and the associated high computational cost of high-resolution simulations. Thus the ideal numerical

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Section: Acceleration On Graphics Processing Unitssupporting

confidence: 94%

“…A full description of the interpolation procedure can be found in [7,13,15]. The coefficients of the polynomial are used to carry out the time stepping algorithm.…”

Section: Description Of the Methodsmentioning

confidence: 99%

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Leapfrog Time-Stepping for Hermite Methods

et al. 2019

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“…The Hermite methods of Goodrich and co-authors were first tailored to graphics processing units by Dye in [10] wherein an implementation for the two dimensional advection equation was proposed. Strategies for three-dimensional equations were later proposed by Vargas and co-authors in [21]. Here we focus on extending the techniques introduced in [21] to develop algorithms for the dissipative and conservative Hermite methods in three space dimensions.…”

Section: An Example In Two Space Dimensionsmentioning

confidence: 99%

“…Strategies for three-dimensional equations were later proposed by Vargas and co-authors in [21]. Here we focus on extending the techniques introduced in [21] to develop algorithms for the dissipative and conservative Hermite methods in three space dimensions.…”

Section: An Example In Two Space Dimensionsmentioning

confidence: 99%

Hermite Methods for the Scalar Wave Equation

Appelö¹,

Hagstrom²,

Vargas³

2018

SIAM J. Sci. Comput.

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Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are presented. Both methods use (m + 1) d degrees of freedom per node for the displacement in d-dimensions; the dissipative and conservative methods achieve orders of accuracy (2m − 1) and 2m, respectively. Stability and error analyses as well as implementation strategies for accelerators are also given.1. Introduction. We construct, analyze, and test arbitrary order dissipative and conservative Hermite methods for the scalar wave equation in a medium with constant speed of sound c. The degrees-of-freedom for Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The dissipative method achieves space-time accuracy of order 2m − 1, while the conservative method has space-time order 2m. Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for c∆t ≤ h, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.Our primary interest in these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type, where boundary conditions are simple to apply. In previous work [7] we considered this type of hybridization of the Hermite methods for first order system proposed in [11] with nodal discontinuous Galerkin (dG) methods. For wave equations in second order form we envision a similar hybridization where the geometry is handled by, for example, our recently developed discontinuous Galerkin methods for wave equations in second order form [1]. Our dG method has the property that, based on the choice of numerical flux, it is either dissipative or conservative.We provide optimal stability and convergence results for both the conservative and dissipative method for one dimensional periodic domains. The analysis for the dissipative method follows the analysis for first order systems [11] but here it is based on the energy of the wave equation v 2 + |∇u| 2 dx. A difference compared to [11] is that we require that the (polynomial) approximation spaces of the velocity, v, and displacement, u to differ by one degree. Its extension to higher space dimensions, however, does not follow in a straightforward way from the Hermite method for first-order systems, requiring a specialized interpolation scheme to achieve order-independent stability at CFL one.The analysis of the conservative method is new and quite different from that of [11]. Additionally, the analysis is done by introducing what we denote conserved variables, this simplifies the analysis conside...

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Hermite Methods in Time

Hagstrom

2020

Lecture Notes in Computational Science and Engineering

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We study time-stepping methods based on Hermite-Birkhoff interpolation. For linear problems, the methods are designed so that their stability characteristics are identical to those of standard implicit Runge–Kutta methods based on Gauss-Legendre quadrature (Hairer and Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems. Springer, New York, 1996). However, in contrast, only a single nonlinear system whose dimension matches that of the original problem must be solved independent of the method order; that is, they are singly implicit. Besides outlining the construction of the methods and establishing some of their basic properties, we carry out illustrative computations both on standard test problems and a spectral discretization of an initial-boundary value problem for the Schrödinger equation.

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GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

Cited by 3 publications

References 14 publications

Leapfrog Time-Stepping for Hermite Methods

Leapfrog Time-Stepping for Hermite Methods

Hermite Methods for the Scalar Wave Equation

Hermite Methods in Time

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