Superconvergence of projection integrators for conservative system

Projection methods are applicable in many fields. It is a natural and practical approach to devise the invariant-preserving schemes for conservative systems. The idea is to project the solution of any underlying numerical scheme onto the manifold determined by the invariant, and this process will be referred to as the projection integrator. Generally, the projection integrator chooses the gradient of invariant as its projection direction and has the same order as the underlying method. In this paper, we propose a different projection direction to construct a new projection integrator whose order is higher than the underlying method. According to this novel direction, we further summarize high-order projection integrators with superconvergence and rigorously prove the truncation error by utilizing the linear integral method as a central tool. Apart from the invariant-preserving property, symmetry is an important geometric property for reversible differential equations. The design and analysis of another high-order projection integrators with symmetry and superconvergence are also presented in this paper. Numerical experiments are provided to verify our theoretical results and illustrate that our proposed projection integrators have superior behaviors in a long time numerical simulation.