We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0...We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0 〈 ε〈〈1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O（ ε^2） and O（1） in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference （CNFD） method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ- as well as the small parameter 0 〈 ε≤1 Based on the error bound, in order to obtain ＇correct＇ numerical solutions in the nonrelativistic limit regime, i.e., 0 〈 ε≤1 , the CNFD method requests the ε-scalability： τ- = O（ε3） and h = O（√ε）. Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time- splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O（ε2） and h = O（1） when 0 〈 ε 〈〈 1. Extensive numerical results are reported to confirm our error estimates.展开更多
In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-pr...In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.展开更多
基金supported by the Ministry of Education of Singapore(Grant No.R146-000-196-112)National Natural Science Foundation of China(Grant No.91430103)
文摘We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation （NLDE） in the nonrelativistic limit regime, involving a small dimensionless parameter 0 〈 ε〈〈1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O（ ε^2） and O（1） in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference （CNFD） method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ- as well as the small parameter 0 〈 ε≤1 Based on the error bound, in order to obtain ＇correct＇ numerical solutions in the nonrelativistic limit regime, i.e., 0 〈 ε≤1 , the CNFD method requests the ε-scalability： τ- = O（ε3） and h = O（√ε）. Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time- splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O（ε2） and h = O（1） when 0 〈 ε 〈〈 1. Extensive numerical results are reported to confirm our error estimates.
基金Project supported by the National Natural Science Foundation of China(No.11071182)
文摘In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Caussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.
