High-order uniformly accurate fully discrete schemes with long time near-conservations for two nonlinear coupled dispersive models in the nonrelativistic regime

Nonlinear coupled dispersive models arise in numerous physical settings, including quantum electrodynamics, nonlinear optics, and plasma dynamics. In the nonrelativistic limit, such systems exhibit rapid temporal oscillations of wavelength $\mathcal{O}({\epsilon }^2)$, controlled by a dimensionless parameter $0<\epsilon \ll 1$. Moreover, owing to the Yukawa-coupled interaction terms, numerical schemes that simultaneously achieve high-order uniform accuracy and approximately preserve the mass and the energy are essential for reliable simulations. In this work, we propose a class of fully discrete schemes for nonlinear coupled dispersive models in the nonrelativistic regime that achieve higher-order uniform accuracy and approximately preserve both invariants over long times. Spatial discretization is based on the Fourier pseudospectral method, and time advancement employs two-scale exponential Runge-Kutta methods. Numerical experiments on the highly oscillatory Klein-Gordon-Dirac and Klein-Gordon-Schrödinger equations in one, two, and three dimensions confirm that the resulting schemes attain the theoretically predicted uniform accuracy and the long-time near-conservation of both invariants.