Physics-based adaptivity of a spectral method for the Vlasov-Poisson
equations based on the asymmetrically-weighted Hermite expansion in velocity
space
We propose a spectral method for the 1D-1V Vlasov-Poisson system where the
discretization in velocity space is based on asymmetrically-weighted Hermite
functions, dynamically adapted via a scaling α and shifting u of the
velocity variable. Specifically, at each time instant an adaptivity criterion
selects new values of α and u based on the numerical solution of the
discrete Vlasov-Poisson system obtained at that time step. Once the new values
of the Hermite parameters α and u are fixed, the Hermite expansion is
updated and the discrete system is further evolved for the next time step. The
procedure is applied iteratively over the desired temporal interval. The key
aspects of the adaptive algorithm are: the map between approximation spaces
associated with different values of the Hermite parameters that preserves total
mass, momentum and energy; and the adaptivity criterion to update α and
u based on physics considerations relating the Hermite parameters to the
average velocity and temperature of each plasma species. For the discretization
of the spatial coordinate, we rely on Fourier functions and use the implicit
midpoint rule for time stepping. The resulting numerical method possesses
intrinsically the property of fluid-kinetic coupling, where the low-order terms
of the expansion are akin to the fluid moments of a macroscopic description of
the plasma, while kinetic physics is retained by adding more spectral terms.
Moreover, the scheme features conservation of total mass, momentum and energy
associated in the discrete, for periodic boundary conditions. A set of
numerical experiments confirms that the adaptive method outperforms the
non-adaptive one in terms of accuracy and stability of the numerical solution