A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization technique. First, the nonlinear Hamiltonian differential equation is transformed into an equivalent reformulation which admits a quadratic energy invariant. For the nonlinear ODE, the reformulation is then discretized using a class of relaxation Runge-Kutta schemes. For the nonlinear PDE, the reformulation is first discretized in the space direction by adopting the Fourier pseudo-spectral discretization which preserves the semi-discrete quadratic conservation laws. Then the semi-discrete system is integrated in the time direction using the explicit relaxation Runge-Kutta method. The obtained method can conserve different quadratic conservation laws to machine accuracy, which improves the numerical stability during long time computations. Besides, the proposed method keeps the same convergence rate of the standard Runge-Kutta scheme when computed with time step size rescaling, and gives a rate of convergence reduced by at most one when computed without step size rescaling. Numerical experiments for several ODEs and PDEs are provided to illustrate the advantages of the proposed algorithms over long time and verify the theoretical analysis.

利用不变能量正交化技术，提出了非线性哈密顿ODE和PDE的不变守恒显式Runge-Kutta格式的统一框架。首先，将非线性哈密顿微分方程转换为等效二次方程，该二次方程允许二次能量不变。对于非线性ODE，然后使用一类弛豫Runge-Kutta方案离散化重构。对于非线性PDE，首先通过采用保留半离散二次守恒定律的傅里叶伪谱离散化，在空间方向上离散化重构。然后，使用显式松弛Runge-Kutta方法在时间方向上将半离散系统集成。所获得的方法可以保存不同的二次守恒律以提高机器精度，这提高了长时间计算过程中的数值稳定性。此外，所提出的方法在进行时间步长缩放的情况下保持与标准Runge-Kutta方案相同的收敛速度，而在不进行步长缩放的情况下进行计算时，收敛速度最多降低1。提供了几种ODE和PDE的数值实验，以说明所提出算法的长期优势，并验证了理论分析。