In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, our key ideas mainly follow the extrapolation/prediction-correction technique and symplectic Runge-Kutta (RK) methods in time combined with the standard Fourier pseudo-spectral method in space. We show that it is uniquely solvable, unconditionally stable and can exactly preserve the momentum of the system. Subsequently, based on the energy quadratization approach and the analogous linearized idea used in the construction of the linear momentum-preserving scheme, the energy-preserving scheme is presented and it is proven to preserve both the discrete mass and quadratic energy. Numerical results are addressed to demonstrate the accuracy and efficiency of the schemes.

翻译：在本文中,为了解决常规化长波方程,分别提出了一系列任意高阶线性保持动力和节能计划。对于节能计划,我们的主要想法主要遵循外推/预测-修正技术和中位龙格-库塔(RK)方法,同时采用标准的Fourier伪光谱空间方法。我们表明,它具有独特的可溶性,无条件稳定,能够完全保持系统的势头。随后,根据能源四分法和构建线性节能计划时使用的类似线性线性概念,提出了节能计划,并证明它既能保护离散质量能源,又能四面形能源。数字结果旨在显示计划的准确性和效率。