Vlasov on GPU (VOG project)

Abstract

Résumé. Ce travail concerne la simulation numérique du modèle de Vlasov-Poissonà l'aide de méthodes semi-Lagrangiennes, sur des architectures GPU. Pour cela, quelques modifications de la méthode traditionnelle ont dûêtre effectuées. Tout d'abord, une reformulation des méthodes semiLagrangiennes est proposée, qui permet de la réécrire sous la forme d'un produit d'une matrice circulante avec le vecteur des inconnues. Ce calcul peutêtre fait efficacement grâce aux routines de FFT. Actuellement, le GPU n'est plus limitéà la simple précision. Néanmoins, la simple précision reste intéressante pour des raisons de performance et de mémoire disponible. Afin de contourner le problème de la simple précision, une méthode de type δf est alors utilisée. Ainsi, un code Vlasov-Poisson GPU permet de simuler et de décrire avec un haut degré de précision (grâceà l'utilisation de reconstructions d'ordreélevé et d'un grand nombre de points de l'espace des phases) des cas tests académiques mais aussi des phénomènes physiques pertinents, comme la simulation des ondes KEEN.Abstract. This work concerns the numerical simulation of the Vlasov-Poisson equation using semiLagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.

Semi-Lagrangian Vlasov simulation on GPUs

“…Thus, obtaining a performant implementation in this setting is very difficult. All approaches to parallelize Eulerian and semi-Lagrangian methods on GPUs have thus focused on different algorithms [25,8,4,30].…”

Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations

“…The advantage of that method is that there is no memory overhead as compared to a finite difference implementation. (Note that the method under consideration in this paper stores the coefficients of the Legendre polynomials up to order which leads to an increased memory consumption but is expected to result in a scheme that is less dissipative; see, e.g., [20].) In the mentioned paper a numerical study of the Van Leer scheme is conducted.…”

Solving the guiding-center model on a regular hexagonal mesh

“…P1 which is the least costly method is also clearly the fastest method. We obtain around 1GFlops (when the efficiency is about 30) which is a correct value with respect to [27] for example. It is followed by the Z9 element, and the Mitchell elements which need also the computation of the mixed derivatives.…”