论文标题
保守系统的明确且实际上不变的方法
An explicit and practically invariants-preserving method for conservative systems
论文作者
论文摘要
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，一种明确的数值策略实际上是为保守系统得出的，该策略是为保守系统而得出的，该方法被称为显式不变性的不变性（EIP）方法。所提出的方法显示出与基础RK方法的顺序相同，而不变式的错误则以$ \ Mathcal {O} \ left（H^{2（p+1）} \ right）的顺序分析，其中$ h $是$ h $是时间步，$ p $代表该方法的顺序。当$ p $适当的较大时，EIP方法实际上是不变的，因为不变的错误可以达到机器的精度。对于ODE和高维PDE，为单个和多个不变性的情况说明了该方法。提出了广泛的数值实验，以验证我们的理论结果，并在长期的数值模拟中证明了所提出方法的优越行为。数值结果表明，四阶EIP方法比标准的四阶RK方法更好地保留了流量的定性特性，并且在实践中比完全隐含的集成符更有效。
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.
