Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equation

In this paper, some linearly implicit modified energy-conserving schemes are proposed for the modified Korteweg-de Vries equation (mKdV). The proposed schemes are based on the recently developed invariant energy quadratization (IEQ) approach and the scalar auxiliary variable (SAV) approach. We first introduce an auxiliary variable to transform the original model into an equivalent system, with a modified energy functional law. Then, the Fourier pseudospectral method is employed for the spatial discretization, and Crank-Nicolson, and Leap-Frog methods are used for the temporal discretization. We analyze the conservation properties, existence and uniqueness and the linear stability of the proposed schemes. The optimal order convergence rate of the semi-discrete scheme and the fully discrete schemes were analyzed, respectively. At last, some numerical examples are presented to illustrate the effectiveness of the proposed schemes.