ERROR ESTIMATES OF NUMERICAL METHODS FOR THE LONG-TIME DYNAMICS OF THE NONLINEAR KLEIN-GORDON EQUATION

Abstract

This work is devoted to the error estimates of numerical methods for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter.The analytical results indicate that the life-span of a smooth solution to this NKGE is at least up to the time at $O(\varepsilon^{-2})$. Different numerical methods are adapted to discretize the problem and rigorous error bounds are established for the long-time dynamics.The numerical methods studied in this work include the finite difference methods, exponential wave integrator methods as well as the time-splitting methods and particular attentions are paid on the error bounds of different numerical methods up to the time $t= T_0/\varepsilon^{\beta}$ with $0 \leq \beta \leq 2$ and $T_0$ fixed. As a by-product, the results are extended to solve an oscillatory NKGE whose solution propagates waves with wavelength at $O(1)$ in space and $O(\varepsilon^{\beta})$ in time. Extensive numerical results are reported to confirm the error bounds and demonstrate that they are sharp.