# Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity

## TL;DR

This paper derives long-time error bounds for finite difference time domain methods applied to the nonlinear Klein-Gordon equation with weak nonlinearity, providing guidelines for mesh size and time step in simulations.

## Contribution

It establishes rigorous error bounds for FDTD methods over long times in the weak nonlinearity regime, with explicit dependence on the small parameter and validation through numerical tests.

## Key findings

- Error bounds valid up to time O(1/ε^β) for 0 ≤ β ≤ 2
- Error bounds depend explicitly on mesh size h, time step τ, and ε
- Numerical results confirm the sharpness of the bounds

## Abstract

We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by $\varepsilon^2$ with $0 <\varepsilon \leq 1$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at $O(1/\varepsilon^{\beta})$ with $0 \le \beta \leq 2$, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $\varepsilon\in (0,1]$, especially in the weak nonlinearity regime when $0 < \varepsilon \ll 1$. Our error bounds indicate that, in order to get ``correct'' numerical solutions up to the time at $O(1/\varepsilon^{\beta})$, the $\varepsilon$-scalability (or meshing strategy) of the FDTD methods should be taken as: $h = O(\varepsilon^{\beta/2})$ and $\tau = O(\varepsilon^{\beta/2})$. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.