Abstract:
In this talk, I will review our recent works on numerical methods and analysis
for solving the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely
proportional to the speed of light. In this regime, the solution is
highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD)
methods and obtain their rigorous error estimates in the
nonrelativistic limit regime by paying particularly attention to how
error bounds depend explicitly on mesh size and time step as
well as the small parameter. Then we consider a numerical method by using
spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives
to discretize the KG equation. Rigorious error estimates
show that the EWI spectral method show much better temporal resolution than the FDTD methods for the KG equation in the nonrelativistic limit regime.
In order to design a multiscale method for the KG equation,
we establish error estimates of FDTD and EWI spectral methods for the nonlinear
Schrodinger equation perturbed with a wave operator. Finally, a multiscale method is presented for discretizing the nonlinear KG equation in the nonrelativistic limit regime based on large-small amplitude wave decompostion. This multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the nonlinear KG equation in the nonrelativistic limite regime. Finally, applications to several high oscillatory dispersive partial differential equations
will be discussed. |