Two-scale integrators with high accuracy and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

TL;DR

This work develops two-scale, uniformly accurate integrators for the nonlinear Klein-Gordon equation in the nonrelativistic limit, leveraging a reformulated two-scale system, Fourier spectral spatial discretization, and multi-stage exponential integrators with symmetry and stiff-order conditions. The resulting S2O3 and S3O4 schemes attain high-order uniform accuracy in time (O(h^3) and O(h^4), respectively) and exhibit near energy conservation over long times, even for large initial data. The authors provide rigorous convergence proofs and long-time energy analysis via modulated Fourier expansions, complemented by 1D and 2D numerical tests that validate uniform behavior in the small parameter $\varepsilon$ and demonstrate practical competitiveness. The framework offers a path toward uniformly accurate, energy-respecting integrators for other highly oscillatory Hamiltonian PDEs.

Abstract

In this paper, we are concerned with two-scale integrators for the non-relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation {in the form of $\partial_{tt}u -\fracΔ{\eps^2} u +\frac{1}{\eps^4}u +\fracλ{\varepsilon^2} f(u)=0$} {has a factor $1/\varepsilon^2$ in front of the nonlinearity which means that this part becomes strong when $\varepsilon$ is small. These} two aspects bring significantly numerical burdens in designing numerical methods. {We propose a class of two-scale integrators which is constructed based on some reformulations to the system, Fourier pseudo-spectral method and exponential integrators.} Two practical integrators up to order three and four are constructed by using some symmetric conditions and the stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the uniform accuracy in time is $\mathcal{O}(h^3)$ and $\mathcal{O}(h^4)$ for the time stepsize $h$. The near energy conservation over long times is also established for the multi-stage integrators by using modulated Fourier expansions.

Figures (13)

• Figure 1: ISV: the log-log plot of the temporal errors $err_u=\frac{\left\Vert u^n-u(\cdot, t _n)\right\Vert_{H^{1}}}{\left\Vert u(\cdot, t _n)\right\Vert_{H^{1}}}$ and $err_v=\frac{\left\Vert v^n-v( \cdot,t _n)\right\Vert_{H^{0}}}{\left\Vert v( \cdot,t _n)\right\Vert_{H^{0}}}$ at $t _n=1$ under different $h$, where $\varepsilon=1/2^k$ with $k=1,2,\ldots,5$.
• Figure 2: S2O3: the log-log plot of the temporal errors $err_u=\frac{\left\Vert u^n-u(\cdot, t _n)\right\Vert_{H^{1}}}{\left\Vert u(\cdot, t _n)\right\Vert_{H^{1}}}$ and $err_v=\frac{\left\Vert v^n-v( \cdot,t _n)\right\Vert_{H^{0}}}{\left\Vert v( \cdot,t _n)\right\Vert_{H^{0}}}$ at $t _n=1$ under different $h$, where $\varepsilon=1/2^k$ with $k=1,2,\ldots,5$.
• Figure 3: S3O4: the log-log plot of the temporal errors $err_u=\frac{\left\Vert u^n-u(\cdot, t _n)\right\Vert_{H^{1}}}{\left\Vert u(\cdot, t _n)\right\Vert_{H^{1}}}$ and $err_v=\frac{\left\Vert v^n-v( \cdot,t _n)\right\Vert_{H^{0}}}{\left\Vert v( \cdot,t _n)\right\Vert_{H^{0}}}$ at $t _n=1$ under different $h$, where $\varepsilon=1/2^k$ with $k=1,2,\ldots,5$.
• Figure 4: ISV: the log-log plot of the temporal errors $err_u=\frac{\left\Vert u^n-u(\cdot, t _n)\right\Vert_{H^{1}}}{\left\Vert u(\cdot, t _n)\right\Vert_{H^{1}}}$ and $err_v=\frac{\left\Vert v^n-v( \cdot,t _n)\right\Vert_{H^{0}}}{\left\Vert v( \cdot,t _n)\right\Vert_{H^{0}}}$ at $t _n=1$ under different $\varepsilon$, where $h=1/2^k$ with $k=6,7,\ldots,10$.
• Figure 5: S2O3: the log-log plot of the temporal errors $err_u=\frac{\left\Vert u^n-u(\cdot, t _n)\right\Vert_{H^{1}}}{\left\Vert u(\cdot, t _n)\right\Vert_{H^{1}}}$ and $err_v=\frac{\left\Vert v^n-v( \cdot,t _n)\right\Vert_{H^{0}}}{\left\Vert v( \cdot,t _n)\right\Vert_{H^{0}}}$ at $t _n=1$ under different $\varepsilon$, where $h=1/2^k$ with $k=6,7,\ldots,10$.

Theorems & Definitions (20)

• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• proof
• Theorem 3.1
• Remark 3.2
• Theorem 3.3