A third-order trigonometric integrator with low regularity for the semilinear Klein-Gordon equation

Abstract

In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed based on the full use of Duhamel's formula and the employment of a twisted function tailored for trigonometric integrals. Robust error analysis is conducted, demonstrating that the proposed scheme achieves third-order accuracy in the energy space under a weak regularity requirement in $H^{1+\max(μ,1)}(\mathbb{T}^d)\times H^{\max(μ,1)}(\mathbb{T}^d)$ with $μ> \frac{d}{2}$. A numerical experiment shows that the proposed third-order low-regularity integrator is much more accurate than some well-known exponential integrators of order three for approximating the Klein-Gordon equation with non-smooth solutions.

Figures (4)

• Figure 1: Spatial error of LRI: the error $serr=\frac{\left\Vert u_n-u(t_n)\right\Vert_{H^{1}}}{\left\Vert u(t_n)\right\Vert_{H^{1}}}+\frac{\left\Vert v_n-v(t_n)\right\Vert_{L^{2}}}{\left\Vert v(t_n)\right\Vert_{L^{2}}}$ with initial data $H^{\theta}(\mathbb{T}) \times H^{\theta-1}(\mathbb{T})$ against different $N_x$.
• Figure 2: Temporal error $err=\frac{\left\Vert u_n-u(t_n)\right\Vert_{H^{1}}}{\left\Vert u(t_n)\right\Vert_{H^{1}}}+\frac{\left\Vert v_n-v(t_n)\right\Vert_{L^{2}}}{\left\Vert v(t_n)\right\Vert_{L^{2}}}$ at $T=1$ with initial data $H^{\theta}(\mathbb{T}) \times H^{\theta-1}(\mathbb{T})$ against $h$ with $h =1/2^k$, where $k=1,2,\ldots,7$.
• Figure 3: Temporal error $err=\frac{\left\Vert u_n-u(t_n)\right\Vert_{H^{1}}}{\left\Vert u(t_n)\right\Vert_{H^{1}}}+\frac{\left\Vert v_n-v(t_n)\right\Vert_{L^{2}}}{\left\Vert v(t_n)\right\Vert_{L^{2}}}$ at $T=1$ with initial data $H^{\theta}(\mathbb{T}) \times H^{\theta-1}(\mathbb{T})$ against $h$ with $h =1/2^k$, where $k=1,2,\ldots,10$.
• Figure 4: Efficiency comparison: $err=\frac{\left\Vert u_n-u(t_n)\right\Vert_{H^{1}}}{\left\Vert u(t_n)\right\Vert_{H^{1}}}+\frac{\left\Vert v_n-v(t_n)\right\Vert_{L^{2}}}{\left\Vert v(t_n)\right\Vert_{L^{2}}}$ at $T=5$ against different CPU time produced by different $h =1/2^k$, where $k=1,2,\ldots,7$.

Theorems & Definitions (16)

• Proposition 2.1
• Proof 1
• Lemma 2.1
• Definition 2.1
• Remark 2.1
• Definition 2.2
• Theorem 3.1
• Lemma 3.1
• Proof 2
• Lemma 3.2