High-order Hamiltonian splitting for the Vlasov–Poisson equations

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• 作者：Fernando Casas ; Nicolas Crouseilles ; Erwan Faou…
• 关键词：Mathematics Subject Classification37M15 Symplectic integrators ; 65M25 Method of characteristics ; 65M12 Stability and convergence of numerical methods ; 65P10 Hamiltonian systems including symplectic integrators ; 65Z05 Applications to physics ; 82D10 Plasmas
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：135
• 期：3
• 页码：769-801
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

We consider the Vlasov–Poisson equation in a Hamiltonian framework and derive new time splitting methods based on the decomposition of the Hamiltonian functional between the kinetic and electric energy. Assuming smoothness of the solutions, we study the order conditions of such methods. It appears that these conditions are of Runge–Kutta–Nyström type. In the one dimensional case, the order conditions can be further simplified, and efficient methods of order 6 with a reduced number of stages can be constructed. In the general case, high-order methods can also be constructed using explicit computations of commutators. Numerical results are performed and show the benefit of using high-order splitting schemes in that context. Complete and self-contained proofs of convergence results and rigorous error estimates are also given.