Furihata and Matsuo proposed in 2010 an energy-conserving scheme for the Zakharov equations, as an application of the discrete variational derivative method (DVDM). This scheme is distinguished from conventional methods (in particular the one devised by Glassey in 1992) in that the invariants are consistent with respect to time, but it has not been sufficiently studied both theoretically and numerically. In this study, we theoretically prove the solvability under the loosest possible assumptions. We also prove the convergence of this DVDM scheme by improving the argument by Glassey. Furthermore, we perform intensive numerical experiments for comparing the above two schemes. It is found that the DVDM scheme is superior in terms of accuracy, but since it is fully-implicit, the linearly-implicit Glassey scheme is better for practical efficiency. In addition, we proposed a way to choose a solution for the first step that would allow Glassey's scheme to work more efficiently.

翻译：Furihata与Matsuo于2010年将离散变分导数法（DVDM）应用于Zakharov方程，提出了一个能量守恒格式。该格式与传统方法（尤其是Glassey于1992年提出的格式）的区别在于其不变量在时间上保持一致，但无论是理论分析还是数值研究均未充分展开。本研究在尽可能宽松的假设条件下，从理论上证明了该格式的可解性。同时，通过改进Glassey的论证方法，证明了该DVDM格式的收敛性。此外，我们开展了大量数值实验以比较上述两种格式。结果表明，DVDM格式在精度方面具有优势，但由于其全隐式特性，线性隐式Glassey格式在实际计算效率上更优。我们还提出了一种为Glassey格式选择初始步解的方法，以提升其计算效率。