Mass and energy conservative numerical methods are proposed for a general system of $N$ strongly coupled nonlinear Schrödinger equations (N-CNLS). Motivated by the structure preserving properties of composition methods, two basic conservative, first and second order time integrators, are developed as seed schemes for the derivation of high order conservative methods. To avoid solving a global nonlinear system, involving all the components of the vector field at each time step, a conservative nonlinear splitting method based on a modified Crank-Nicolson scheme is proposed. Conservation of the mass for each component and total energy is formally proved for the semi-discrete primal formulation of the Local Discontinuous Galerkin (LDG) method and for the fully discrete methods. Since the proposed splitting scheme is independent of the spatial discretization, conservation of the same invariants is also obtained for other symmetric discontinuous Galerkin discretizations. Conservation and accuracy of the discrete invariants; and, spatial and temporal convergence are numerically validated on a series of benchmark (2/3)-CNLS systems. Using a special projector operator, the approximated initial energy of the system is shown, numerically, to convergence with order $\mathcal{O}\left(h^{2p+2}\right)$ when polynomials of degree $p$ are used.

提出了质量和能量保守数值方法的一般系统。 $ñ$强耦合非线性Schrödinger方程（N-CNLS）。受合成方法的结构保留特性的影响，开发了两种基本的保守的一阶和二阶时间积分器，作为推导高阶保守方法的种子方案。为避免求解包含每个时间步向量场所有分量的全局非线性系统，提出了一种基于改进的Crank-Nicolson方案的保守非线性分裂方法。对于局部不连续伽勒金（LDG）方法的半离散原始公式和完全离散方法，已正式证明了每种成分和总能量的守恒性。由于建议的分割方案与空间离散无关，对于其他对称的不连续Galerkin离散化，也可以获得相同不变式的守恒。离散不变式的守恒性和准确性；并且，在一系列基准（2/3）-CNLS系统上通过数值验证了空间和时间收敛。使用特殊的投影仪操作员，以数字的形式显示了系统的近似初始能量，从而有序收敛$\mathcal{Ø}（H^{\mathrm{2个}p+\mathrm{2个}}）$ 当多项式 $p$ 被使用。