arXiv subjects

Caoyi Liu

Publications and source records attributed to Caoyi Liu.

2 recordsLinked to original sources

arXiv2026

A Uniformly Accurate Multiscale Time Integrator for the Klein-Gordon-Schr\"odinger Equations in the Nonrelativistic Regime via Simplified Transmission Conditions

We propose a novel and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Klein-Gordon-Schr\"odinger (KGS) equations with a dimensionless parameter epsilon in (0,1], where epsilon is inversely proportional to the speed of light. The proposed MTI-FP method is rigorously proved to achieve uniform first-order accuracy in time in the nonrelativistic regime, i.e., as epsilon->0. In this regime, the solution of the KGS equations exhibits temporal oscillations with an O(epsilon^2)-wavelength, imposing stringent resolution requirements on classical numerical methods. The uniformly accurate MTI-FP method is built upon two key points: (i) a multiscale decomposition by frequency in each time interval with simplified transmission conditions, and (ii) an exponential integrator for temporal discretization combined with the Fourier pseudospectral method for spatial discretization. Using the energy method and mathematical induction, we rigorously establish two independent error bounds in H^1-norm at O(h^{m0-1} + tau^2/epsilon^2) and O(h^{m0-1} + epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution. These estimates imply that the MTI-FP method converges uniformly and optimally in space, and uniformly in time at O(tau) with respect to epsilon in (0,1]. Furthermore, by incorporating a linear interpolation of the micro-variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any t>0. Consequently, the proposed MTI-FP method has a super-resolution property in time from the perspective of Shannon sampling theory. Ample numerical experiments are provided to validate the error estimates and to demonstrate the super-resolution property. Finally, the method is applied to numerically investigate the convergence rates of the KGS equations to different limiting models.

math.NA↗

arXiv2026

A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.

math.NA↗