无界区域上广义Benjamin-Ono方程的高阶守恒格式
High order conservative schemes for the generalized Benjamin-Ono equation in the unbounded domain
摘要：
本文对广义Benjamin-Ono（BO）方程提出了一类新的在时间上具有任意高阶精度的质量或能量守恒的数值格式。空间离散化采用准谱方法，其基函数为有理基函数，可以通过快速傅里叶变换（FFT）实现，计算量为$\mathcal{O}( N\log(N))$。通过将空间离散系统转化为不同的等价形式，使得空间半离散系统的质量或能量在连续时间流下都能精确保持。结合辛龙格-库塔法，无论有无标量辅助变量变换，都可以分别构造出具有任意高阶时间精度的能量守恒和质量守恒的全离散格式.数值结果表明，所提出的格式具有守恒性，并且比非守恒（蛙跳）格式具有更高的精度和稳定性。
英文摘要：
This paper proposes a new class of mass or energy conservative numerical schemes for the generalized Benjamin-Ono (BO) equation on the whole real line with arbitrarily high-order accuracy in time. The spatial discretization is achieved by the pseudo-spectral method with the rational basis functions, which can be implemented by the Fast Fourier transform (FFT) with the computational cost $\mathcal{O}( N\log(N))$. By reformulating the spatial discretized system into the different equivalent forms, either the spatial semi-discretized mass or energy can be preserved exactly under the continuous time flow. Combined with the symplectic Runge-Kutta, with or without the scalar auxiliary variable reformulation, the fully discrete energy or mass conservative scheme can be constructed with arbitrarily high-order temporal accuracy, respectively. Our numerical results show the conservation of the proposed schemes, and also the superior accuracy and stability to the non-conservative (Leap-frog) scheme.
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