A new framework for the construction and analysis of exponential wave integrators for the Zakharov system

The main challenge in the analysis of numerical methods for the Zakharov system (ZS) originates from the presence of derivatives in the nonlinearity. In this paper, we present a novel reformulation of the ZS, which allows us to construct second-order time symmetric methods and higher-order numerical methods for the ZS even with generalized nonlinear terms. By considering exponential wave integrators (EWIs) for this reformulation, a new time symmetric EWI is formulated and its properties are rigorously studied. The proposed method is proved to have two conservation laws at the discrete level. The second-order convergence in time is rigorously shown under a time-step restriction that is independent of the spatial discretization. Moreover, by the strategy presented in this paper, higher-order methods are obtained for the ZS with generalized nonlinear terms. Numerical explorations confirm the theoretical results and superiority of the proposed integrators.