Vlasov Equations

Mathematics / Research groups / Numerical Analysis and Scientific Computing / Archive / Vlasov Equations

Project summary

  • Project name: Splitting methods for the Vlasov-Poisson and Vlasov-Maxwell equations
  • FWF project id: P25346
  • University: University of Innsbruck, Department of Mathematics
  • Field: numerical analysis, applied mathematics, plasma physics
  • Keywords: abstract evolution equations, convergence analysis, splitting methods, high-order methods, discontinuous Galerkin, Vlasov-Maxwell equation

Short project description

The Vlasov-Poisson and Vlasov-Maxwell equations are the most fundamental description of a (collisionless) plasma. The equations describe the evolution of a particle-probability distribution in 3+3 dimensional phase space coupled to an electromagnetic field. The difficulties in obtaining a numerical solution of those equations are summarized in the following three statements:

  1. Due to the six-dimensional phase space the amount of memory required to store the interpolation is proportional to the sixth power of the number of grid points.
  2. The Vlasov equation is stiff (i.e. the time step size is limited by the CFL condition)
  3. The coupling to the Maxwell/Poisson equation makes the system highly non-linear.

A numerical scheme based on Strang splitting has been proposed that translates the basis functions of some interpolation space and projects the translated basis function back onto the proper subspace.

Various interpolation schemes for the above mentioned algorithm have also been investigated. It has been found that the discontinuous Galerkin method is extremely competitive performance wise, while most of its desirable features (such as locality) remain.

Therefore the aim of this project is to:

  1. supply an in-depth numerical analysis of Strang splitting for Vlasov-type equations;
  2. extend the achieved results to higher-order splitting methods;
  3. provide a convergence analysis of the fully discrete problems (using discontinuous Galerkin in space);
  4. extend the previous results to higher-order methods in space.

To implement higher-order splitting methods we have to evaluate the force term at certain intermediate steps. We will develop a strategy that leads to computationally efficient schemes.

One page abstract (english, pdf)
Einseitige Kurzfassung (deutsch, pdf)

Publications

Papers

2016

Lukas Einkemmer, Mayya Tokman, John Loffeld
On the performance of exponential integrators for problems in magnetohydrodynamics
Journal of Computational Physics, Vol. 330, pp. 550-565 (ScienceDirect, arXiv)

Nicolas Crouseilles, Lukas Einkemmer, Erwan Faou
An asymptotic preserving scheme for the relativistic Vlasov-Maxwell equations in the classical limit
Computer Physics Communications, Vol. 209, pp. 13-26 (ScienceDirect, arXiv)

Lukas Einkemmer
Structure preserving numerical methods for the Vlasov equation
Oberwolfach Reports No. 18/2016 (pdf, arXiv)

L. Einkemmer
High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code
Computer Physics Communications, Vol. 202, pp. 326-336 (ScienceDirect, arXiv)

L. Einkemmer
A resistive magnetohydrodynamics solver using modern C++ and the Boost library
Computer Physics Communications, Vol. 206, pp. 69-77 (ScienceDirect, arXiv)

2015

L. Einkemmer, A. Ostermann
A splitting approach for the Kadomtsev-Petviashvili equation
Journal of Computational Physics 2015, Vol. 299, pp. 716-730 (ScienceDirect, arXiv)

L. Einkemmer, A. Ostermann
Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions
SIAM Journal on Scientific Computing 2015, Vol. 37, No. 3, pp. A1577-A1592 (SIAM, arXiv)

L. Einkemmer, A. Ostermann
On the error propagation of semi-Lagrange and Fourier methods for advection problems
Computers & Mathematics with Applications 2015, Vol. 69, No. 3, pp.170-179 (ScienceDirect - OpenAccess)

N. Crouseilles, L. Einkemmer, E. Faou
A Hamiltonian splitting for the Vlasov-Maxwell system
Journal of Computational Physics 2015, Vol. 283, pp. 224-240 (ScienceDirect, arXiv)

2014

L. Einkemmer, M. Wiesenberger
A conservative discontinuous Galerkin scheme for the 2D incompressible Navier-Stokes equations
Computer Physics Communications 2014, Vol. 185, Issue 11, p. 2865–2873 (ScienceDirect, arXiv)

L. Einkemmer, A. Ostermann
A comparison of triple jump and Suzuki fractals for obtaining high order from an almost symmetric Strang splitting scheme
Oberwolfach Reports No. 14/2014 (pdf)

L. Einkemmer, A. Ostermann
A strategy to suppress recurrence in grid-based Vlasov solvers
The European Physical Journal D 2014, Vol. 68, p. 197 (Springer, arXiv)

L. Einkemmer, A. Ostermann
An almost symmetric Strang splitting scheme for the construction of high order composition methods
Journal of Computational and Applied Mathematics 2014, Vol. 271, pp. 307-318 (ScienceDirect - OpenAccess)

L. Einkemmer, A. Ostermann
An almost symmetric Strang splitting scheme for nonlinear evolution equations
Computers & Mathematics with Applications 2014, Vol. 67, Issue 12, pp. 2144-2157 (ScienceDirect - Open Access)

L. Einkemmer, A. Ostermann
Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov-Poisson equations
SIAM Journal on Numerical Analysis 2014, Vol. 52, No. 2, pp. 757-778 (SIAM, arXiv)

L. Einkemmer, A. Ostermann
Convergence analysis of Strang splitting for Vlasov-type equations
SIAM Journal on Numerical Analysis 2014, Vol. 52, No. 1, pp. 140-155 (SIAM, arXiv)

Preprints

L. Einkemmer
Evaluation of the Intel Xeon Phi and NVIDIA K80 as accelerators for two-dimensional panel codes
Preprint (arXiv)

Contact

Lukas Einkemmer
Lukas.Einkemmer@uibk.ac.at
Technikerstraße 13, office: 706
A-6020 Innsbruck, Austria

Alexander Ostermann
Alexander.Ostermann@uibk.ac.at
Technikerstraße 13, office: 701
A-6020 Innsbruck, Austria

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