Mathematical analysis and numerical comparison of energy-conservative schemes for the Zakharov equations

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.