The convergence treatment of particle methods for multidimensional, periodic vlasov-poisson systems
Havlak, Karl Jason
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In this thesis, we have extended and refined the theory obtained in [SIAM J. Numer. Anal., 26 (1989), pp. 249-288] for the Vlasov-Poisson Cauchy problem to include multidimensional periodic systems. This work directly generalizes the analysis by G.-H. Cottet and P. A. Raviart [SIAM J. Numer. Anal., 21 (1984), pp. 52-76], and earlier work by H. Neunzert and J. Wick [Numer. Math., 21 (1973), pp. 234-243], on one-dimensional periodic Vlasov-Poisson systems. We have also relaxed the requirement that the problem data possess compact support with respect to velocity. This is accomplished by the use of a weighted discrete LP-norm which involves discrete velocity moments of the errors in the Hamiltonian trajectories. Such an idea was motivated by the clever analysis by P. L. Lions and B. Perthame [Invent. Math., 105 (1991 ), pp. 415-430] treating global existence of solutions to three-dimensional Vlasov-Poisson systems. These authors have shown that control of certain velocity moments of the distribution for all time leads to the existence of a global classical solution. Analogously, we can show that if we can control the weighted discrete LP -errors in the particle trajectories with respect to time, then we can get globally uniform estimates for the errors.