Semi-Lagrangian Vlasov simulation on GPUs

Abstract

In this paper, our goal is to efficiently solve the Vlasov equation on GPUs. A semi-Lagrangian discontinuous Galerkin scheme is used for the discretization. Such kinetic computations are extremely expensive due to the high-dimensional phase space. The SLDG code abstracts the number of dimensions and uses a shared code base for both GPU and CPU based simulations. We investigate the performance of the implementation on a range of both Tesla (V100, Titan V, K80) and consumer (GTX 1080 Ti) GPUs. Our implementation is typically able to achieve a performance of approximately 470 GB/s on a single GPU and 1600 GB/s on four V100 GPUs connected via NVLink. This results in a speed up of about a factor of ten (comparing a single GPU with a dual socket Intel Xeon Gold node) and approximately a factor of 35 (comparing a single node with and without GPUs). In addition, we investigate the effect of single precision computation on the performance of the SLDG code and demonstrate that a template based dimension independent implementation can achieve good performance regardless of the dimensionality of the problem.

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

“…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.…”

“…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.…”

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

“…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.…”

“…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.…”

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

“…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].…”