Variations on Hermite Methods for Wave Propagation

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

doi:10.4208/cicp.260915.281116a

Cited by 4 publications

(7 citation statements)

References 37 publications

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“…A significant feature of the method's stability is that the result is independent of order. We refer the reader to [5,7,15] for further details on the methods.…”

Section: Overview Of Hermite Methodsmentioning

confidence: 99%

GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

Vargas

Chan

Hagstrom

et al. 2017

Lecture Notes in Computational Science and Engineering

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The Hermite methods of Goodrich, Hagstrom, and Lorenz (2006) use Hermite interpolation to construct high order numerical methods for hyperbolic initial value problems. The structure of the method has several favorable features for parallel computing. In this work, we propose algorithms that take advantage of the many-core architecture of Graphics Processing Units. The algorithm exploits the compact stencil of Hermite methods and uses data structures that allow for efficient data load and stores. Additionally the highly localized evolution operator of Hermite methods allows us to combine multi-stage time-stepping methods within the new algorithms incurring minimal accesses of global memory. Using a scalar linear wave equation, we study the algorithm by considering Hermite interpolation and evolution as individual kernels and alternatively combined them into a monolithic kernel. For both approaches we demonstrate strategies to increase performance. Our numerical experiments show that although a two kernel approach allows for better performance on the hardware, a monolithic kernel can offer a comparable time to solution with less global memory usage.

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“…A significant feature of the method's stability is that the result is independent of order. We refer the reader to [5,7,15] for further details on the methods.…”

Section: Overview Of Hermite Methodsmentioning

confidence: 99%

GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

Vargas

Chan

Hagstrom

et al. 2017

Lecture Notes in Computational Science and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…Notably, the coefficients of the polynomial are the approximation of the function value and first 2m + 1 derivatives at the node x j . As demonstrated in [15] this results in the following system…”

Section: Description Of the Methodsmentioning

confidence: 73%

“…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%

“…To compliment the experiments, we present comparisons with the classic Hermite method (Dual Hermite method as referred to in [15]). As a model equation we chose the following wave system…”

Section: Numerical Experiments In One-dimensionmentioning

confidence: 99%

“…Adaptive variants of Hermite methods have been introduced by means of padaptivity in [4], and preliminary work on h-adaptivity has been presented in [2]. Variations of the method which do not require a dual grid have been introduced in [15], and a flux conservative Hermite method has been introduced in [8].…”

mentioning

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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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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Variations on Hermite Methods for Wave Propagation

Cited by 4 publications

References 37 publications

GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

GPU Acceleration of Hermite Methods for the Simulation of Wave Propagation

Leapfrog Time-Stepping for Hermite Methods

Hermite Methods in Time

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