Sparse Grids for the Vlasov–Poisson Equation

Abstract

The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equation. Due to the high dimensionality and the development of small structures the numerical solution is quite challenging. For two or four dimensional Vlasov problems, semi-Lagrangian solvers have been successfully applied. Introducing a sparse grid, the number of grid points can be reduced in higher dimensions. In this paper, we present a semi-Lagrangian Vlasov-Poisson solver on a tensor product of two sparse grids. In order to defeat the problem of poor representation of Gaussians on the sparse grid, we introduce a multiplicative delta-f method and separate a Gaussian part that is then handled analytically. In the semi-Lagrangian setting, we have to evaluate the hierarchical surplus on each mesh point. This interpolation step is quite expensive on a sparse grid due to the global nature of the basis functions. In our method, we use an operator splitting so that the advection steps boil down to a number of one dimensional interpolation problems. With this structure in mind we devise an evaluation algorithm with constant instead of logarithmic complexity per grid point. Results are shown for standard test cases and in four dimensional phase space the results are compared to a full-grid solution and a solution on the four dimensional sparse grid.

Vlasov methods in space physics and astrophysics

“… 2004 ; Thomas 2016 ). Another possibility here is to globally reduce the number of grid points by introducing a sparse grid representation, where the grid may be uneven with respect to Cartesian coordinates, while remaining static during runtime (Kormann and Sonnendrücker 2016 ; Guo and Cheng 2016 ). This is sometimes referred to as a sparse grid representation.…”

“…5.2 . This method is not to be mixed to the static sparse grid methods (Kormann and Sonnendrücker 2016 ; Guo and Cheng 2016 ) that are fundamentally dimension reduction techniques, similar to the low-rank approximations. In plasmas, the magnetic field makes the particles gyrate while the electric field causes them to accelerate and drift.…”

Semi-Lagrangian Vlasov simulation on GPUs

“…We will also discuss some aspects that are pertinent to the present work in the next section. Let us also mention that recently the use of dimension reduction techniques, such as low-rank approximations, have been explored [22,23,13,12,18,15].…”

Sparse grid discontinuous Galerkin methods for the Vlasov-Maxwell system

“…we have B = 0. This is a classical test example that been previously computed by many algorithms [16,15,25,26]. Example 3.2 We consider the 2D2V Landau damping with two spatial variables x 1 and x 2 , and two velocity variables ξ 1 and ξ 2…”