An explicit numerical strategy that practically preserves invariants is derived for conservative systems by combining an explicit high-order Runge-Kutta (RK) scheme with a simple modification of the standard projection approach, which is named the explicit invariants-preserving (EIP) method. The proposed approach is shown to have the same order as the underlying RK method, while the error of invariants is analyzed in the order of $\mathcal{O}\left(h^{2(p+1)}\right),$ where $h$ is the time step and $p$ represents the order of the method. When $p$ is appropriately large, the EIP method is practically invariants-conserving because the error of invariants can reach the machine accuracy. The method is illustrated for the cases of single and multiple invariants, with regard to both ODEs and high-dimensional PDEs. Extensive numerical experiments are presented to verify our theoretical results and demonstrate the superior behaviors of the proposed method in a long time numerical simulation. Numerical results suggest that the fourth-order EIP method preserves much better the qualitative properties of the flow than the standard fourth-order RK method and it is more efficient in practice than the fully implicit integrators.

中文翻译：

---

一种用于保守系统的显式且实际上保持不变的方法

通过将显式高阶 Runge-Kutta (RK) 方案与标准投影方法的简单修改相结合，为保守系统推导出一种实际上保留不变量的显式数值策略，该方法称为显式不变量保留 (EIP) 方法。所提出的方法被证明具有与底层 RK 方法相同的顺序，而不变量的误差按 $\mathcal{O}\left(h^{2(p+1)}\right) 的顺序进行分析， $ 其中 $h$ 是时间步长，$p$ 表示方法的顺序。当 $p$ 适当大时，EIP 方法实际上是不变量守恒的，因为不变量的误差可以达到机器精度。针对 ODE 和高维 PDE 的单个和多个不变量的情况说明了该方法。大量的数值实验被提出来验证我们的理论结果，并在长时间的数值模拟中证明了所提出的方法的优越行为。数值结果表明，四阶 EIP 方法比标准四阶 RK 方法更好地保留了流的定性特性，并且在实践中比完全隐式积分器更有效。