Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

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 ε 2 with 0<ε≤1 a dimensionless parameter. When 0 < ε ≪ 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(ε). 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/ε β ) with 0 ≤ β ≤ 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 τ as well as the small parameter ε ∈ (0,1], especially in the weak nonlinearity regime when 0 < ε ≪ 1. Our error bounds indicate that, in order to get "correct" numerical solutions up to the time at O(1/ε β ), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h =O(ε β/2 ) and τ =O(ε β/2 ). As a by-product, our results can indicate error bounds and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE which is obtained from the case of weak nonlinearity by a rescaling in time, while its solution propagates waves with wavelength at O(1) in space and O(ε β ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity

“…Compared with the commonly used standard FDTD methods [30], the results can attain higher order accuracy in space for a given mesh size or improve the spatial resolution capacity, that is, it needs less grid points while maintaining the same accuracy.…”

“…We remark here that in the approximation of the first step vale, in order to uniformly bound v 1 ∈ X M for ∈ (0, 1], k −p and k 2 −2p are replaced by sin(k −p ) and ksin(k −2p ), respectively [30].…”

“…The implicit 4cFD 8 can be solved via either a direct solver or an iterative method. The semi‐implicit 4cFD 9 can be explicitly updated in the Fourier space with O ( M ln M ) computational cost per time step [30, 37]. For other semi‐implicit and explicit schemes, we leave them to readers who are interested in.…”

“…The result indicates that the 4cFD methods have better spatial resolution capacity than the FDTD methods [30]. Also, it is useful for choosing mesh size and time step such that the numerical results are trustable.…”

“…Since the life‐span of the solution to the NKGE (1.1) is up to the time at O ( ϵ − p ), it is natural to establish error bounds of the numerical method for the NKGE (1.1) up to the time at O ( ϵ − p ) instead of O (1). In our recent work [30], four explicit/semi‐implicit/implicit conservative/nonconservative FDTD methods for the NKGE (1.1) with a cubic nonlinearity, that is, p = 2, have been proposed and analyzed. The error estimates of the FDTD methods indicate that, in order to obtain ‘correct’ numerical solution of the NKGE (1.1) with a cubic nonlinearity, the ϵ −scalability is: h = O ( ϵ ) and τ = O ( ϵ ).…”