Numerical instabilities in multistep symplectic methods triggered via krein collisions

Discrete variational integrators comprise an interesting class of symplectic integrators that can be applied to any Lagrangian system. In plasma physics, such integrators have been applied, for example, to the guiding center (GC) system [1]. However, these integrators may be multistep methods, and thus can suffer from parasitic instabilities [2]. It is known that instabilities in continuous Hamiltonian and symplectic systems are only triggered under specific conditions, namely via the Krein collision in which two eigenmodes with different signs of action resonate. We show that since symplectic integrators conserve the same symplectic form as the underlying continuous system, numerical spectral stability can only be lost by the same Krein collision mechanism. This formalism is used to explain the onset of parasitic instabilities observed in a particular variational integrator for GC dynamics [1]. We also explore the simultaneous application of Krein’s theorem and Dahlquist’s equivalence theorem to multistep symplectic methods and examine whether these can produce particularly robust numerical stability under certain spectral conditions.

[1] H. Qin and X. Guan, Phys. Rev. Letters 100, 035006 (2008).

[2] C. L. Ellison, Ph.D. thesis, 2006.