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A generalized Fourier–Hermite semi-discretization for the Vlasov–Poisson equation is introduced. The formulation of the method includes as special cases the symmetrically-weighted and asymmetrically-weighted Fourier–Hermite methods from the literature. The numerical scheme is formulated as a weighted Galerkin method with two separate scaling parameters for the Hermite polynomial and the exponential part of the new basis functions. Exact formulas for the error in mass, momentum, and energy conservation of the method depending on the parameters are devised and L2stability is discussed. The numerical experiments show that an optimal choice of the additional parameter in the generalized method can yield improved accuracy compared to the existing methods, but also reveal the distinct stability properties of the symmetrically-weighted method.

Abstract: We present a memory-efficient and parallel framework for finite element operator application implemented in the generic open-source library deal.II. Instead of assembling a sparse matrix and using it for matrix–vector products, the operation is applied by cell-wise quadrature. The evaluation of shape functions is implemented with a sum-factorization approach. Our implementation is parallelized on three levels to exploit modern supercomputer architecture in an optimal way: MPI over remote nodes, thread parallelization with dynamic task scheduling within the nodes, and explicit vectorization for utilizing processors’ vector units. Special data structures are designed for high performance and to keep the memory requirements to a minimum. The framework handles adaptively refined meshes and systems of partial differential equations. We provide performance tests for both linear and nonlinear PDEs which show that our cell-based implementation is faster than sparse matrix–vector products for polynomial order two and higher on hexahedral elements and yields ten times higher Gflops rates. [Copyright &y& Elsevier]

This paper describes our approach to developing a simulation software application for the fully kinetic 6D-Vlasov equation, which will be used to explore physics beyond the reduced gyrokinetic model. Simulating the fully kinetic Vlasov equation requires efficient utilization of compute and storage capabilities due to the high dimensionality of the problem. In addition, the implementation needs to be extensible regarding the physical model and flexible regarding the hardware for production runs. We start on the algorithmic background to simulate the 6-D Vlasov equation using a semi-Lagrangian algorithm. The performance portable software stack, which enables production runs on pure CPU as well as AMD or Nvidia GPU accelerated nodes, is presented. The extensibility of our implementation is guaranteed through the described software architecture of the main kernel, which achieves a memory bandwidth of almost 500 GB/s on a V100 Nvidia GPU and around 100 GB/s on an Intel Xeon Gold CPU using a single code base. We provide performance data on multiple node-level architectures discussing utilized and further available hardware capabilities. Finally, the network communication bottleneck of 6-D grid-based algorithms is quantified. A verification of physics beyond gyrokinetic theory, for the example of ion Bernstein waves, concludes the work. • Performance portable implementation of a semi-Lagrangian algorithm for full kinetics. • Software architecture for Lagrange interpolation stencils using design patterns. • Node level performance analysis for OpenMP, HIP and CUDA using a single code base. • Quantification of the network communication bottleneck for 6D distributed grids. [ABSTRACT FROM AUTHOR]