An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the...An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.展开更多
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
文摘An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.
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.
