The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

a b s t r a c tWe give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger, (linear) time-dependent Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

Abstract.We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.Mathematics Subject Classification. 65P10, 65L06. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods preserve arbitrary quadratic invariants [4]. However, no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [1]; the linear systemẋ = x,ẏ = y,ż = −2z, with invariant xyz, provides an example. (Preservation would require R(h) 2 R(−2h) ≡ 1, where R is the stability function of the method 4 ; but this requires R(h) = e h , which is impossible [6].) This result does not rule out, however, the existence of RK methods that preserve particular (as opposed to arbitrary) invariants. Since the invariant does not appear in the RK method, this will require some special relationship between the invariant and the vector field. Such a relationship does exist in the case of the energy invariant of canonical Hamiltonian systems: see [3,5] on energy-preserving B-series. In this article we show that for any polynomial Hamiltonian function, there exists an RK method of any order that preserves it.The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equatioṅthe AVF method is the map x → x defined by

We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painlevé equation. We also discuss how Manin transformations arise in Kahan discretizations of certain vector fields.

A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy when applied to quadratic Hamiltonian vector fields. Here we generalize Kahan's method to cubic resp. higher degree polynomial vector fields and show that the resulting discretization also preserves modified versions of the measure and energy when applied to cubic resp. higher degree polynomial Hamiltonian vector fields