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In this paper, a class of arbitrarily high-order linear momentum-preservingand energy-preserving schemes are proposed, respectively, for solving theregularized long-wave equation. For the momentum-preserving scheme, the keyidea is based on the extrapolation/prediction-correction technique and thesymplectic Runge-Kutta method in time, together with the standard Fourierpseudo-spectral method in space. We show that the scheme is linear, high-order,unconditionally stable and preserves the discrete momentum of the system. Forthe energy-preserving scheme, it is mainly based on the energy quadratizationapproach and the analogous linearized strategy used in the construction of thelinear momentum-preserving scheme. The proposed scheme is linear, high-orderand can preserve a discrete quadratic energy exactly. Numerical results areaddressed to demonstrate the accuracy and efficiency of the proposed scheme.

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation