Related papers: Long time error analysis of finite difference time…
We present the fourth-order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE), while the nonlinearity strength is characterized by $\varepsilon^p$ with a constant $p…
We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 <…
We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is…
In this paper, we derive the improved uniform error bounds for the long-time dynamics of the $d$-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by…
We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by…
We compare the long-time error bounds and spatial resolution of finite difference methods with different spatial discretizations for the Dirac equation with small electromagnetic potentials characterized by $\varepsilon \in (0, 1]$ a…
Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in (0,1]$, which is inversely proportional to the speed…
We propose and analyze two regularized finite difference methods for the logarithmic Klein-Gordon equation (LogKGE). Due to the blowup phenomena caused by the logarithmic nonlinearity of the LogKGE, it is difficult to construct numerical…
We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless…
We present and analyze two regularized finite difference methods which preserve energy of the logarithmic Klein-Gordon equation (LogKGE). In order to avoid singularity caused by the logarithmic nonlinearity of the LogKGE, we propose a…
We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit…
We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with highly oscillatory initial data in the form of a modulated plane wave. In this regime, the solution exhibits rapid oscillations in both time and space,…
An improved uniform error bound at $O\left(h^m+\varepsilon^2 \tau^2\right)$ is established in $H^{\alpha/2}$-norm for the long-time dynamics of the nonlinear space fractional Klein-Gordon equation (NSFKGE). A second-order exponential wave…
We consider a nonlinear Klein--Gordon equation in the nonrelativistic limit regime with initial data in the form of a modulated highly oscillatory exponential. In this regime of a small scaling parameter $\varepsilon$, the solution exhibits…
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$…
Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where the nonlinearity strength is characterized by a dimensionless parameter…
The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger equation in an explicitly iterative process. However, the method requires the spatial grid…
We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter $0< \varepsilon \ll…
In this paper, we propose a neural multiscale decomposition method (NeuralMD) for solving the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in(0,1]$ from the relativistic regime to the nonrelativistic…
The finite difference time domain (FDTD) method has been successfully applied to obtain energies and wave functions for two electrons in a quantum dot modeled by a three dimensional harmonic potential. The FDTD method uses the…