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A multiscale time integrator Fourier pseudospectral (MTI-FP) method isproposed and analyzed for solving the Klein-Gordon-Schr\"{o}dinger (KGS)equations in the nonrelativistic limit regime with a dimensionless parameter$0<\varepsilon\le1$ which is inversely proportional to the speed of light. Infact, the solution to the KGS equations propagates waves with wavelength at$O(\varepsilon^2)$ and $O(1)$ in time and space, respectively, when$0<\varepsilon\ll 1$, which brings significantly numerical burdens in practicalcomputation. The MTI-FP method is designed by adapting a multiscaledecomposition by frequency to the solution at each time step and applying theFourier pseudospectral discretization and exponential wave integrators forspatial and temporal derivatives, respectively. We rigorously establish twoindependent error bounds for the MTI-FP at $O(\tau^2/\varepsilon^2+h^{m_0})$and $O(\varepsilon^2+h^{m_0})$ for $\varepsilon\in(0,1]$ with $\tau$ time stepsize, $h$ mesh size and $m_0\ge 4$ an integer depending on the regularity ofthe solution, which imply that the MTI-FP converges uniformly and optimally inspace with exponential convergence rate if the solution is smooth, anduniformly in time with linear convergence rate at $O(\tau)$ for$\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at$O(\tau^2)$ in the regime when either $\varepsilon=O(1)$ or $0<\varepsilon\le\tau$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) ofthe MTI-FP is $\tau=O(1)$ and $h=O(1)$ for $0<\varepsilon\ll 1$, which issignificantly better than classical methods. Numerical results demonstrate thatour error bounds are optimal and sharp. Finally, the MTI-FP method is appliedto study numerically convergence rates of the KGS equations to the limitingmodels in the nonrelativistic limit regime.

numerische mathematik

A uniformly accurate (UA) multiscale time integrator Fourier  pseoduspectral method for the Klein-Gordon-Schrodinger equations in the  nonrelativistic limit regime