Abstract. While developing a new semi-Lagrangian solver, the gap between a linear Landau run in 1D×1D and a 5D gyrokinetic simulation in toroidal geometry is quite huge. Intermediate test cases are welcome for testing the code. A new fully two-dimensional conservative semi-Lagrangian (CSL) method is presented here and is validated on 2D polar geometries. We consider here as building block, a 2D guiding-center type equation on an annulus and apply it on two test cases. First, we revisit a 2D test case previously done with a PIC approach [18] and detail the boundary conditions. Second, we consider a 4D drift-kinetic slab simulation (see [10]). In both cases, the new method appears to be a good alternative to deal with this type of models since it improves the lack of mass conservation of the standard semi-Lagrangian (BSL) method.
Abstract. We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov-Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [ACS12], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss-Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge-Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).
We develop adaptive numerical schemes for the Vlasov equation by combining discontinuous Galerkin discretisation, multiresolution analysis and semi-Lagrangian time integration. We implement a tree based structure in order to achieve adaptivity. Both multi-wavelets and discontinuous Galerkin rely on a local polynomial basis. The schemes are tested and validated using Vlasov-Poisson equations for plasma physics and astrophysics.
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