Improved uniform error bounds for long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation with weak nonlinearity

TL;DR

This work studies the long-time dynamics of the high-dimensional nonlinear space fractional sine-Gordon equation with weak nonlinearity ($0<\varepsilon\le 1$) in $d=2,3$, and derives improved uniform error bounds up to time $T_\varepsilon=T/\varepsilon^2$ using a second-order time-splitting method coupled with a Fourier pseudospectral spatial discretization. A regularity compensation oscillation (RCO) technique is introduced to make the error explicitly depend on $\varepsilon$, yielding $O(\varepsilon^2\tau^2)$ for semi-discretization and $O(h^m+\varepsilon^2\tau^2)$ for fully discrete schemes. The study extends the TSFP framework to complex and oscillatory complex NSFSGE, with corresponding error bounds, and validates the theory through extensive 2D/3D numerical experiments, including soliton dynamics and collisions. The results illuminate how fractional order $\alpha$ influences dynamics and demonstrate the practical viability of the method for long-time simulations of nonlinear fractional wave models.

Abstract

In this paper, we derive the improved uniform error bounds for the long-time dynamics of the $d$-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by $\varepsilon^2$ where $0<\varepsilon \le 1$ is a dimensionless parameter. The second-order time-splitting method is applied to the temporal discretization and the Fourier pseudo-spectral method is used for the spatial discretization. To obtain the explicit relation between the numerical errors and the parameter $\varepsilon$, we introduce the regularity compensation oscillation technique to the convergence analysis of fractional models. Then we establish the improved uniform error bounds $O\left(\varepsilon^2 τ^2\right)$ for the semi-discretization scheme and $O\left(h^m+\varepsilon^2 τ^2\right)$ for the full-discretization scheme up to the long time at $O(1/\varepsilon^2)$. Further, we extend the time-splitting Fourier pseudo-spectral method to the complex NSFSGE as well as the oscillatory complex NSFSGE, and the improved uniform error bounds for them are also given. Finally, extensive numerical examples in two-dimension or three-dimension are provided to support the theoretical analysis. The differences in dynamic behaviors between the fractional sine-Gordon equation and classical sine-Gordon equation are also discussed.

Figures (18)

• Figure 1: Long-time spatial errors for the NSFSGE in 2D at $t=1/\varepsilon^2$ with different $\alpha$.
• Figure 2: Long-time temporal errors for the NSFSGE in 2D at $t=1/\varepsilon^2$ with different $\alpha$.
• Figure 3: Long-time temporal errors for the discrete energy of the NSFSGE in 2D at $t=1/\varepsilon^2$ with different $\alpha$.
• Figure 4: Long-time spatial errors for the NSFSGE in 3D at $t=1/\varepsilon^2$ with different $\alpha$.
• Figure 5: Long-time temporal errors for the NSFSGE in 3D at $t=1/\varepsilon^2$ with different $\alpha$.

Theorems & Definitions (12)

• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2
• proof
• Remark 3.1
• Remark 3.2