Geometric integration of ODEs using multiple quadratic auxiliary variables

Abstract

We present a novel method for solving ordinary differential equations (ODEs) while preserving all polynomial first integrals. The method is essentially a symplectic Runge-Kutta method applied to a reformulated version of the ODE under study and is illustrated through a number of examples including Hamiltonian ODEs, a Nambu system and the Toda Lattice. When applied to certain Hamiltonian ODEs, the proposed method yields the averaged vector field method as a special case.

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

“…Based on the basic principle of the structure-preserving method where the numerical method should preserve the intrinsic properties of the original problems as much as possible, it is valuable to expect that the high-order mass and energy-preserving discretizations for the QZS (1.1) will produce richer information on the continuous system. Very recently, inspired by the ideas of the invariant energy quadratization (IEQ) approach [53], a new class of high-order accurate energypreserving methods are proposed in [21,33]. Especially, the term "quadratic auxiliary variable (QAV) approach" was coined by Gong et al in [21].…”