Comment on 'Symplectic Integration of Magnetic Systems': A Proof that the Boris Algorithm is Not Variational

Authors: C. Leland Ellison, J.W. Burby, and H. Qin

Abstract:  The Boris algorithm is a popular technique for the numerical time advance of charged particles interacting with electric and magnetic fields according to the Lorentz force law [1, 2, 3, 4]. Its popularity stems from simple implementation, rapid iteration, and excellent long-term numerical fidelity [1, 5].  Excellent long-term behavior strongly suggests the numerical dynamics exhibit conservation laws analogous to those governing the continuous Lorentz force system [6]. Without conserved quantities to constrain the numerical dynamics, algorithms typically dissipate or accumulate important observables such as energy and momentum over long periods of simulated time [6]. Identication of the conservative properties of an algorithm is important for establishing rigorous expectations on the long-term behavior; energy-preserving, symplectic, and volume-preserving methods each have particular implications for the qualitative numerical behavior [6, 7, 8, 9, 10, 11].
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Submitted to: Journal of Computational Physics
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