Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we establish a family of symplectic integrators for a classof high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.Then we apply the symplectic Euler method to the Hamiltonian system.It is demonstrated that the scheme not only preserves symplectic geometry structureof the original system, but also does not require to resolve coupled nonlinearalgebraic equations which is different from the general implicit symplectic schemes.The linear stability of the symplectic Euler scheme and the errors of the numericalsolutions are investigated. It shows that the semi-explicit scheme is conditionallystable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical testssuggest that the symplectic integrators are more effective than non-symplectic ones,such as backward Euler integrators.