Uniform error bounds of time-splitting spectral methods for the long-time dynamics of the nonlinear Klein–Gordon equation with weak nonlinearity

Abstract

We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein–Gordon equation (NKGE) with weak power-type nonlinearity and O ( 1 ) O(1) initial data, while the nonlinearity strength is characterized by ε p \varepsilon ^{p} with a constant p ∈ N + p \in \mathbb {N}^+ and a dimensionless parameter ε ∈ ( 0 , 1 ] \varepsilon \in (0, 1] , for the long-time dynamics up to the time at O ( ε − β ) O(\varepsilon ^{-\beta }) with 0 ≤ β ≤ p 0 \leq \beta \leq p . In fact, when 0 > ε ≪ 1 0 > \varepsilon \ll 1 , the problem is equivalent to the long-time dynamics of NKGE with small initial data and O ( 1 ) O(1) nonlinearity strength, while the amplitude of the initial data (and the solution) is at O ( ε ) O(\varepsilon ) . By reformulating the NKGE into a relativistic nonlinear Schrödinger equation, we adapt the TSFP method to discretize it numerically. By using the method of mathematical induction to bound the numerical solution, we prove uniform error bounds at O ( h m + ε p − β τ 2 ) O(h^{m}+\varepsilon ^{p-\beta }\tau ^2) of the TSFP method with h h mesh size, τ \tau time step and m ≥ 2 m\ge 2 depending on the regularity of the solution. The error bounds are uniformly accurate for the long-time simulation up to the time at O ( ε − β ) O(\varepsilon ^{-\beta }) and uniformly valid for ε ∈ ( 0 , 1 ] \varepsilon \in (0,1] . Especially, the error bounds are uniformly at the second-order rate for the large time step τ = O ( ε − ( p − β ) / 2 ) \tau = O(\varepsilon ^{-(p-\beta )/2}) in the parameter regime 0 ≤ β > p 0\le \beta >p . Numerical results are reported to confirm our error bounds in the long-time regime. Finally, the TSFP method and its error bounds are extended to a highly oscillatory complex NKGE which propagates waves with wavelength at O ( 1 ) O(1) in space and O ( ε β ) O(\varepsilon ^{\beta }) in time and wave velocity at O ( ε − β ) O(\varepsilon ^{-\beta }) .