The minimal stage, energy preserving Runge–Kutta method for polynomial Hamiltonian systems is the averaged vector field method

Abstract

No Runge–Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the averaged vector field (AVF) method can be interpreted as a Runge–Kutta method whose weights b i b_i and abscissae c i c_i represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge–Kutta scheme must in fact be identical to the AVF scheme.