We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation per time step. We show that some existing spatial discretizations, including the popular Fourier spectral method, are in fact energy-conserving if one considers the appropriate form of the energy density. We develop a new relaxation-type approach for conserving multiple nonlinear functionals that is more efficient and robust for the NLS equation compared to the existing multiple-relaxation approach. The accuracy and efficiency of the new schemes is demonstrated on test problems for both the focusing and defocusing NLS.

翻译：我们提出了一类用于非线性薛定谔方程的数值方法，该方法能同时守恒质量和能量，在空间和时间上具有任意高阶精度，且每个时间步仅需求解一个标量代数方程。我们证明，若考虑能量密度的适当形式，包括流行的傅里叶谱方法在内的某些现有空间离散格式实际上也是能量守恒的。针对非线性薛定谔方程，我们发展了一种新的松弛型方法来守恒多个非线性泛函，与现有的多重松弛方法相比，该方法更高效且更稳健。通过聚焦型和非聚焦型非线性薛定谔方程的测试算例，验证了新格式的精度与效率。