Solving Vlasov-Poisson system with an adaptive Hermite spectral method

TL;DR

This paper introduces an adaptive Hermite spectral method for solving the Vlasov-Poisson system, utilizing a frequency indicator and a fast projection operator to improve resolution and preserve invariants.

Contribution

The paper develops a novel adaptive Hermite spectral method with a frequency indicator and an efficient projection operator for the Vlasov-Poisson system.

Findings

Validated the method with 1D1V and 2D2V numerical experiments.

Demonstrated improved resolution and efficiency in simulations.

Preserved key physical invariants during computations.

Abstract

We propose an adaptive Hermite spectral method for the Vlasov-Poisson system based on a recently developed frequency indicator that measures the contribution of the high-order expansion coefficients. Precisely, the symmetrically weighted Hermite basis with a scaling factor is utilized to approximate the distribution function to satisfy the increasing resolution requirement, which, for example, is induced by filamentation. To implement the scaling adjustment, a fast conservative projection operator is constructed in two steps. The first step is to formulate the projection as a constrained optimization problem to preserve key invariants, including mass, momentum, energy, and the $L^2$ norm of the distribution function. The second step is an ODE-based approximation developed to compute the updated expansion coefficients with linear complexity. Numerical experiments with 1D1V and 2D2V settings validate the feasibility and efficiency of this proposed adaptive Hermite method.