An efficient scalar auxiliary variable (SAV) spectral Petrov–Galerkin approximation for nonlinear Hamiltonian systems

Abstract

In this paper, a new scalar auxiliary variable (SAV) spectral Petrov–Galerkin approximation is proposed and studied for nonlinear Hamiltonian systems. The new algorithm is built upon the SAV approach and the spectral Petrov–Galerkin (SPG) method, which enjoys both the advantages of SAV and SPG methods such as the properties of energy preserving and high order accuracy. A rigorous theoretical analysis is provided to show that the proposed algorithm is well-posed and convergent. Furthermore, the conservation properties are investigated. In particular, we prove that SAV-SPG method preserves the modified energy exactly and maintains the symplectic structure up to a spectral accuracy. Numerical experiments have been conducted to verify the efficacy of our algorithm.