High order conservative schemes for the generalized Benjamin-Ono equation in the unbounded domain
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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 𝒪( Nlog(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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