A new type of energy-preserving integrators for quasi-bi-Hamiltonian systems

Abstract

Energy-preserving algorithms, as one of the core research areas in numerical ordinary differential equations, have achieved great success by many methods such as symplectic methods and discrete gradient methods. This paper considers the numerical integration of quasi-bi-Hamiltonian systems, which, as a generalization of bi-Hamiltonian systems, can be expressed in two distinct ways: ${\dot{y}} = P _ { 1 } ( y ) \nabla H _ { 2 }(y) = \frac{1}{\rho (y)}P _ { 2 } ( y ) \nabla H _ { 1 }(y)$. The quasi-bi-Hamiltonian systems have two Hamiltonians $H_1(y)$ and $H_2(y)$. Conventional discrete gradient methods can only preserve one Hamiltonian at a time. In this paper, based on discrete gradient and projection, new energy-preserving integrators that can preserve the two Hamiltonians simultaneously are proposed. They show better qualitative behaviours than traditional discrete gradient methods do. Numerical integrations of Hénon-Heiles type systems and the Korteweg-de Vries (KdV) equation are conducted to show the effectiveness of the new integrators in comparison with traditional discrete gradient methods.