Grid-free weighted particle method applied to the Vlasov–Poisson equation

Abstract

We study a grid-free particle method based on following the evolution of the characteristics of the Vlasov–Poisson system, and we show that it converges for smooth enough initial data. This method is built as a combination of well-studied building blocks—mainly time integration and integral quadratures—and allows to obtain arbitrarily high orders. By making use of the Non-Uniform Fast Fourier Transform, the overall computational complexity is $$ {\mathcal {O}}(P \log P + K^d \log K^d) $$ , where $$ P $$ is the total number of particles and where we only keep the Fourier modes $$ k \in ({\mathbb {Z}}^d)^* $$ such that $$ k_1^2 + \dots + k_d^2 \le K^2 $$ . Some numerical results are given for the Vlasov–Poisson system in the one-dimensional case.