A decoupled, linearly implicit and high-order structure-preserving scheme for Euler-Poincaré equations

It is challenging to develop high-order structure-preserving finite difference schemes for the modified two-component Euler-Poincare equations due to the nonlinear terms and high-order derivative terms. To overcome the difficulties, we introduce a bi-variate function and carefully choose the intermediate average variable in the temporal discretization. Then, we obtain a decoupled and linearly implicit scheme. It is shown that the fully-discrete scheme can keep both the discrete mass and energy conserved. And the fully-discrete scheme has fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction. Several numerical examples are given to confirm the theoretical results.