Time Symmetric and Asymptotic Preserving Exponential Wave Integrators for the Quantum Zakharov System

The primary difficulty in analysis of numerical methods for the quantum Zakharov system (QZS) stems from the inclusion of derivative terms within its nonlinearity. In this work, we present a novel formulation of the QZS which allows us to construct second-order time symmetric and asymptotic preserving methods. Based on this new formulation, a new time symmetric exponential wave integrator (EWI) is formulated and its properties are rigorously studied. The proposed method is proved to have two conservation laws in the discrete level. The second order convergence in time is rigorously shown independent of the spatial discretization mesh size and is maintained in both the strong quantum regime ϑ=1 and the classical regime ϑ=0. Moreover, the new scheme exhibits asymptotic preserving properties, converging uniformly to the classical Zakharov system as ϑ→0. Additionally, the methodology introduced in this paper facilitates the derivation of higher-order time-symmetric methods for the QZS equipped with generalized nonlinearities. Numerical explorations confirm the theoretical results and superiorities of the proposed integrators.