A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.