TY - GEN

T1 - Sparse grids for the vlasov–poisson equation

AU - Kormann, Katharina

AU - Sonnendrücker, Eric

N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2016.

PY - 2016

Y1 - 2016

N2 - The Vlasov–Poisson equation models the evolution of a plasma in an external or self-consistent electric field. Themodel 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.

AB - The Vlasov–Poisson equation models the evolution of a plasma in an external or self-consistent electric field. Themodel 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.

UR - http://www.scopus.com/inward/record.url?scp=84962532179&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-28262-6_7

DO - 10.1007/978-3-319-28262-6_7

M3 - Conference contribution

AN - SCOPUS:84962532179

SN - 9783319282602

T3 - Lecture Notes in Computational Science and Engineering

SP - 163

EP - 190

BT - Sparse Grids and Applications, 2014

A2 - Pflüger, Dirk

A2 - Garcke, Jochen

PB - Springer Verlag

T2 - 3rd Workshop on Sparse Grids and Applications, SGA 2014

Y2 - 1 September 2014 through 5 September 2014

ER -