Daniele Funaro

Gianmarco Manzini, Daniele Funaro, Gian Luca Delzanno

We prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term, that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in details for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence, under suitable regularity assumptions on the exact solution.

Lorella Fatone, Daniele Funaro, Gianmarco Manzini

The Vlasov-Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approx- imated by a Semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along an high-order accurate representation of the local characteristic lines. The time-advancing technique is based on Taylor developments that can be, in principle, of any order of accuracy, or by coupling the phase space discretiza- tion with high-order accurate Backward Differentiation Formulas (BDF) as in the method-of-lines framework. At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine excellent features of spectral approximations with high-order time integration. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.

Lorella Fatone, Daniele Funaro, Gianmarco Manzini

In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on the standard two-stream instability benchmark problem. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

Lorella Fatone, Daniele Funaro

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.

Daniele Funaro

Can we train a machine to detect if another machine has understood a concept? In principle, this is possible by conducting tests on the subject of that concept. However we want this procedure to be done by avoiding direct questions. In other words, we would like to isolate the absolute meaning of an abstract idea by putting it into a class of equivalence, hence without adopting straight definitions or showing how this idea "works" in practice. We discuss the metaphysical implications hidden in the above question, with the aim of providing a plausible reference framework.