Semi-Lagrangian Vlasov simulation on GPUs

Einkemmer, Lukas

doi:10.1016/j.cpc.2020.107351

Cited by 17 publications

(29 citation statements)

References 36 publications

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“…The semi-discrete equation is evaluated in only a few lines of code by utilizing CUDA wrappers with NumPy-like data arrays, allowing tensor-product index ordering in a simple routine. This approach does not outperform a custom implementation with CUDA code [9], yet it has the advantage for beginners of simplicity. In the case of a hyperbolic problem, if the sign of the advection speed is constant during a problem then the sign arrays should be computed prior to the main loop.…”

Section: Discussionmentioning

confidence: 99%

“…Additionally, the DG literature pushes the boundaries of the method with hybridizable [5], semi-Lagrangian [6], superconvergent [7], and space-time [8] innovations. Studies show impressive performance and scaling of DG-type methods on GPUs [9]. Yet there seems to be a gap in the recent literature, namely an easy-to-understand description of a vanilla DG method on a GPU.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of tensor-product discontinous Galerkin operators for Vlasov-Poisson simulations and GPU implementation on Python

Crews¹

2022

Preprint

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The discontinuous Galerkin (DG) finite element method is conservative, lends itself well to parallelization, and is high-order accurate due to its close affinity with the theory of quadrature and orthogonal polynomials. When applied with an orthogonal discretization (i.e. a rectilinear grid) the DG method may be efficiently implemented on a GPU in just a few lines of high-level language such as Python. This work demonstrates such an implementation by writing the DG semi-discrete equation in a tensor-product form and then computing the products using open source GPU libraries. The results are illustrated by simulating a problem in plasma physics, namely an instability in the magnetized Vlasov-Poisson system. Further, as DG is closely related to spectral methods through its orthogonal basis it is possible to calculate a transformation to an alternative set of global eigenfunctions for purposes of analysis or to perform additional operations. This transformation is also posed as a tensor product and may be GPU-accelerated. In this work a Fourier series is computed for example (although this does not beat discrete Fourier transform), and is used to solve the Poisson part of the Vlasov-Poisson system to O(∆x n+1/2 )accuracy.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of tensor-product discontinous Galerkin operators for Vlasov-Poisson simulations and GPU implementation on Python

Crews¹

2022

Preprint

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“…Roughly speaking, we can think of the regularized Fourier multiplier as a numerical trick in order to avoid treating the 0th k-frequency separately. A similar idea is used for example in [10,18]. Notice that the computation of the terms involving b is done in the physical space.…”

Section: Full Discretization With Fft and Wenomentioning

confidence: 99%

“…There is a flourishing literature about use of GPUs in order to accelerate scientific computations, e.g. [6,10,12,13,28]. Reducing the simulation time is of great importance when we aim to model physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

An exponential integrator/WENO discretization for sonic-boom simulation on modern computer hardware

Einkemmer¹,

Ostermann²,

Residori³

2021

Preprint

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Recently a splitting approach has been presented for the simulation of sonicboom propagation. Splitting methods allow one to divide complicated partial differential equations into simpler parts that are solved by specifically tailored numerical schemes. The present work proposes a second order exponential integrator for the numerical solution of sonic-boom propagation modelled through a dispersive equation with Burgers' nonlinearity. The linear terms are efficiently solved in frequency space through FFT, while the nonlinear terms are efficiently solved by a WENO scheme. The numerical method is designed to be highly parallelisable and therefore takes full advantage of modern computer hardware. The new approach also improves the accuracy compared to the splitting method and it reduces oscillations. The enclosed numerical results illustrate that parallelisation on a CPU results in a speedup of 22 times faster than the straightforward sequential version. The GPU implementation further accelerates the runtime by a factor 3, which improves to 5 when single precision is used instead of double precision.

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“…Therefore, methods which are highly parallelizable and work well on these new computer architectures are needed to take advantage of their computational power. A significant body of research has been accumulated in recent years that considers numerical methods that are well suited for such systems (see, e.g., [15,16,21,27,28]). More specifically, in the context of exponential integrators we refer to [17,18].…”

Section: Introductionmentioning

confidence: 99%

An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

Caliari,

Einkemmer,

Moriggl

et al. 2020

Preprint

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Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.

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Semi-Lagrangian Vlasov simulation on GPUs

Cited by 17 publications

References 36 publications

Analysis of tensor-product discontinous Galerkin operators for Vlasov-Poisson simulations and GPU implementation on Python

Analysis of tensor-product discontinous Galerkin operators for Vlasov-Poisson simulations and GPU implementation on Python

An exponential integrator/WENO discretization for sonic-boom simulation on modern computer hardware

An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

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