Quantitative evaluations of stability and convergence for solutions of semilinear Klein--Gordon equation

Abstract

We perform some simulations of the semilinear Klein--Gordon equation with a power-law nonlinear term and propose each of the quantitative evaluation methods for the stability and convergence of numerical solutions. We also investigate each of the thresholds in the methods by varying the amplitude of the initial value and the mass, and propose appropriate values.

Figures (4)

• Figure 1: $\phi$ with $A=2$, $m=3.9$ to $4.2$, and $8000$ grids. The top-left panel is for $m=3.9$, the top-center one is for $m=4.0$, the top-right one is for $m=4.1$, and the bottom one is for $m=4.2$. The vibration appears to occur at $t\geq 500$ for $m=4.0$ and at $t\geq 700$ for $m=4.1$.
• Figure 2: $\phi$ with $A=3$, $m=7.6$ to $8.2$, and $8000$ grids. The top-left panel is for $m=7.6$, the top-center one is for $m=7.7$, the top-right one is for $m=7.8$, the center-left one is for $m=7.9$, the center one is for $m=8.0$, the center-right one is for $m=8.1$, and the bottom one is for $m=8.2$. The vibration appears to occur at $t\geq 900$ for $m=7.8$, at $t\geq 300$ for $m=7.9$, and at $t\geq400$ for $m=8.0$.
• Figure 3: Relative errors between $\phi$ with $8000$ grids and $\phi$ with other grid numbers when $A=2$ and $m=3.9$ to $4.2$. The vertical axis is $CV_g$ and the horizontal axis is time. The top-left panel is for $m=3.9$, the top-center one is for $m=4.0$, the top-right one is for $m=4.1$, and the bottom one is for $m=4.2$. The convergence seems not satisfied at either $t\geq 400$ for $m=4.0$ or $t\geq 350$ for $m=4.1$.
• Figure 4: Relative errors between $\phi$ with $8000$ grids and $\phi$ with other grid numbers when $A=3$ and $m=7.6$ to $8.2$. The top-left panel is for $m=7.6$, the top-center one is for $m=7.7$, the top-right one is for $m=7.8$, the center-left one is for $m=7.9$, the center one is for $m=8.0$, the center-right one is for $m=8.1$, and the bottom one is for $m=8.2$. The convergence seems not satisfied at $t\geq 200$ for $m=7.7$, at $t\geq 250$ for $m=7.8$, at $t\geq 250$ for $m=7.9$, at $t\geq 400$ for $m=8.0$, or at $t\geq 250$ for $m=8.1$.