Long time error analysis of the fourth-order compact finite difference methods for the nonlinearKlein-Gordonequation with weak nonlinearity

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 epsilon(p)with a constantp is an element of N(+)and a dimensionless parameter epsilon is an element of (0, 1]. Based on analytical results of the life-span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the time atO(epsilon(-p)). We pay particular attention to how error bounds depend explicitly on the mesh sizehand time step tau as well as the small parameter epsilon is an element of (0, 1], which indicate that, in order to obtain 'correct' numerical solutions up to the time atO(epsilon(-p)), the epsilon-scalability (or meshing strategy requirement) of the 4cFD methods should be taken as:h = O(epsilon(p/4))and tau = O(epsilon(p/2)). It has better spatial resolution capacity than the classical second order central difference methods. By a rescaling in time, it is equivalent to an oscillatory NKGE whose solution propagates waves with wavelength atO(1)in space andO(epsilon(p))in time. It is straightforward to get the error bounds of the oscillatory NKGE in the fixed time. Finally, numerical results are provided to confirm our theoretical analysis.

Automated summary

This study introduces fourth-order compact finite difference discretizations for the long time dynamics of the nonlinear Klein-Gordon equation and analyzes the error bounds of the method. The research finds that in order to obtain correct numerical solutions, the mesh size and time step of the method need to be consistent with the nonlinearity strength. The method has better spatial resolution capacity compared to traditional central difference methods.