Vlasov simulations on an adaptive phase-space grid

Abstract

The numerical resolution of the Vlasov equation is usually performed by particle methods (PIC) which consist in approximating the plasma by a finite number of particles. The trajectories of these particles are computed from the characteristic curves given by the Vlasov equation, whereas selfconsistent fields are computed on a mesh of the physical space. This method allows to obtain satisfying results with a relatively small number of particles. However, it is well known that, in some cases, the numerical noise inherent to the particle method becomes too important to have an accurate description of the distribution function in phase space. To remedy to this problem, methods discretizing the Vlasov equation on a mesh of phase space have been proposed [2,5,6].The major drawback of Vlasov methods using a uniform and fixed mesh is that their numerical cost is high, which makes them rather inefficient when the dimension of phase-space grows. For this reason we are investigating a method using an adaptive mesh. The adaptive method is overlayed to a classical semi-Lagrangian method which is based on the conservation of the distribution function along particle trajectories. The phase-space grid is updated using a multiresolution technique.The model we consider throughout this paper is the nonrelativistic Vlasov equation coupled self-consistently with Poisson's equation. It readsthe self electric field E is computed from Poisson's equationsThe magnetic field is external and considered to be known. In the present work, we have chosen to introduce a phase-space mesh which can be refined or derefined adaptively in time. For this purpose, we use a technique based on multiresolution analysis which is in the same spirit as the methods developed in particular by S. Bertoluzza [1], A. Cohen et al. [3] and M. Griebel and F. Koster [4]. We represent the distribution function on a wavelet basis at different scales. We can then compress it by eliminating coefficients which are small and accordingly remove the associated mesh points. Another specific feature of our method is that we use an advection in physical and velocity space forward in time to predict the useful grid points for the next time step, rather than restrict ourselves to the neighboring points. This enables us to use a much larger time step, as in the semi-Lagrangian method the time step is not limited by a Courant condition. Once the new mesh is predicted, the semi-Lagrangian methodology is used to compute the new values of the distribution function at the predicted mesh points, using an interpolation based on the wavelet decomposition of the old distribution function. The mesh is then refined again by performing a wavelet transform, and eliminating the points associated to small coefficients. This paper is organized as follows : we shall first present the context of multiresolution analysis for adaptive simulations, then describe the adaptive semi-Lagrangian method and validate our method on typical simulation in beam physics.

Kinetic properties of particle-in-cell simulations compromised by Monte Carlo collisions

“…When the number of real space dimensions is greater than one, it is usually conceded that Monte Carlo methods enjoy an advantage over finite difference methods, essentially because a finite difference method entails an explicit computational mesh spanning all of the real and velocity space dimensions that are represented, and this leads to difficulties when the number of dimensions is large. Adaptive meshing techniques have produced improvements in the performance of finite difference solutions of the Vlasov equation [13], but it is presently not clear that these methods can be extended to the collisional case. So it seems that particle methods are likely to remain the direct Boltzmann solver of choice for most problems in the immediate future, unless there are important innovations in some other type of direct solver.…”

Moments conservation in adaptive Vlasov solver

“…Then, we perform the semi-Lagrangian algorithm on the predicted points and finally compute the wavelet decomposition of f n+1 to get the new set of important details and the corresponding adaptive grid (see [3] for more details on the algorithm).…”

“…Adaptivity, that is refinement or derefinement of the mesh, is based on a multiresolution analysis (section 1, see [3] for more details). In the present work, we focus on moment conservation for the Vlasov equation during the thresholding step.…”

A Hermite-Type Adaptive Semi-Lagrangian Scheme

“…They include wavelet techniques (Gutnic et al, 2004;Gutnic et al, 2005), the moving mesh method , and hierarchical finite element decomposition (Campos Pinto and Mehrenberger, 2004;. One main advantage of the latter method is that the underlying dyadic partition of cells allows for an efficient parallelization.…”