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

TL;DR

This paper conducts simulations of the semilinear Klein--Gordon equation to quantitatively evaluate the stability and convergence of numerical solutions, analyzing thresholds based on initial amplitude and mass.

Contribution

It introduces specific quantitative evaluation methods for stability and convergence and identifies optimal thresholds by varying initial conditions.

Findings

Proposed new methods for stability and convergence evaluation.

Identified thresholds for initial amplitude and mass affecting solution stability.

Validated evaluation methods through numerical simulations.

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.

Tables

Table 1. Table 1: Time when S ​ V 8000 > ε s SV_{8000}>\varepsilon_{\mathrm{s}} for A = 2 A=2 and m = 3.9 m=3.9 to 4.2 4.2 , and for A = 3 A=3 and m = 7.6 m=7.6 to 8.2 8.2 . The left table is for A = 2 A=2 and the right one is for A = 3 A=3 . ○ \bigcirc means that S ​ V 8000 ≤ ε s SV_{8000}\leq\varepsilon_{\mathrm{s}} is always satisfied for 0 ≤ t ≤ 1000 0\leq t\leq 1000 . The values represent the times when S ​ V 8000 > ε s SV_{8000}>\varepsilon_{\mathrm{s}} .

$A=2$
	$3.9$	$4.0$	$4.1$	$4.2$
$0.001$	$○$	$417$	$368$	$○$
$0.06$	$○$	$474$	$414$	$○$
$0.12$	$○$	$488$	$425$	$○$
$0.18$	$○$	$496$	$433$	$○$
$0.24$	$○$	$497$	$558$	$○$
$0.30$	$○$	$504$	$○$	$○$

Table 2. Table 2: Time when D ​ C ​ V 2000 > ε c DCV_{2000}>\varepsilon_{\mathrm{c}} for A = 2 A=2 and m = 3.9 m=3.9 to 4.2 4.2 , and for A = 3 A=3 and m = 7.6 m=7.6 to 8.2 8.2 . The left table is for A = 2 A=2 and the right one is for A = 3 A=3 . ○ \bigcirc means that D ​ C ​ V 2000 ≤ ε c DCV_{2000}\leq\varepsilon_{\mathrm{c}} is always satisfied at 0 ≤ t ≤ 1000 0\leq t\leq 1000 . The values represent the times when D ​ C ​ V 2000 > ε c DCV_{2000}>\varepsilon_{\mathrm{c}} .

$A=2$
	$3.9$	$4.0$	$4.1$	$4.2$
$0.1$	$333$	$423$	$353$	$450$
$0.15$	$○$	$431$	$354$	$○$
$0.2$	$○$	$431$	$354$	$○$
$0.25$	$○$	$431$	$363$	$○$
$0.3$	$○$	$439$	$363$	$○$
$0.35$	$○$	$439$	$365$	$○$
$0.4$	$○$	$440$	$365$	$○$

Equations

$$-\frac{1}{c^2}{\partial }_t^2\varphi +{\delta }^{ij}({\partial }_i{\partial }_j\varphi )-\frac{c^2{m}^2}{{\hslash }^2}\varphi =\lambda |\varphi {|}^{p-1}\varphi ,$$

$$\mathcal{H}$$

$$\frac{1}{c}{\partial }_t\varphi$$

$$\frac{1}{c}{\partial }_t\psi$$

$$\mathcal{H}{}_{(\mathbf{k})}^{(\ell )}$$

$$\frac{{\varphi }_{(\mathbf{k})}^{(\ell +1)}-{\varphi }_{(\mathbf{k})}^{(\ell )}}{c\Delta t}:=\frac{1}{2L_0}({\psi }_{(\mathbf{k})}^{(\ell +1)}+{\psi }_{(\mathbf{k})}^{(\ell )}),$$

$$\frac{{\psi }_{(\mathbf{k})}^{(\ell +1)}-{\psi }_{(\mathbf{k})}^{(\ell )}}{c\Delta t}:=L_0(-\frac{\lambda }{p+1}\frac{|{\varphi }_{(\mathbf{k})}^{(\ell +1)}{|}^{p+1}-|{\varphi }_{(\mathbf{k})}^{(\ell )}{|}^{p+1}}{{\varphi }_{(\mathbf{k})}^{(\ell +1)}-{\varphi }_{(\mathbf{k})}^{(\ell )}}+\frac{{\delta }^{ij}{\hat{\delta }}_i^{⟨1⟩}{\hat{\delta }}_j^{⟨1⟩}({\varphi }_{(\mathbf{k})}^{(\ell +1)}+{\varphi }_{(\mathbf{k})}^{(\ell )})}{2}-\frac{c^2{m}^2({\varphi }_{(\mathbf{k})}^{(\ell +1)}+{\varphi }_{(\mathbf{k})}^{(\ell )})}{2{\hslash }^2}),$$

$${\hat{\delta }}_i^{⟨1⟩}{u}_{(\mathbf{k})}^{(\ell )}:=\frac{u_{(k_1,\dots ,k_i+1,\dots ,k_n)}^{(\ell )}-u_{(k_1,\dots ,k_i-1,\dots ,k_n)}^{(\ell )}}{2\Delta x^i}.$$

$$\frac{|{\varphi }_{(\mathbf{k})}^{(\ell +1)}{|}^{p+1}-|{\varphi }_{(\mathbf{k})}^{(\ell )}{|}^{p+1}}{{\varphi }_{(\mathbf{k})}^{(\ell +1)}-{\varphi }_{(\mathbf{k})}^{(\ell )}}$$

$$d{\varphi }_{(\mathbf{k})}^{(\ell )}:={\hat{s}}_i^{+}{\varphi }_{(\mathbf{k})}^{(\ell )}-{\varphi }_{(\mathbf{k})}^{(\ell )}$$

$$({\hat{s}}_i^{+}d{\varphi }_{(\mathbf{k})}^{(\ell )})d{\varphi }_{(\mathbf{k})}^{(\ell )}<0$$

$$C{V}_g(t)$$

$$DC{V}_g(t)$$

Full text

1 Introduction

Many natural phenomena are expressed by (nonlinear) hyperbolic equations. We are strongly interested in the behavior of the asymptotic solutions of the equations in time. In addition, the properties of the solutions should be changed in a curved spacetime since the differential operator is affected by the curvature of spacetime (e.g.[1]). We adopt the Klein–Gordon equation as the hyperbolic equation since it can be applied to a curved spacetime. Some analytical results of the equation in the de Sitter spacetime, which is one of the curved spacetimes, have been reported [2, 3]. Regarding the numerical study of the equation, we have reported the numerical solutions of the equation in the de Sitter spacetime using the structure-preserving scheme [4], suggested some discrete equations constructed using the structure-preserving scheme [5], and investigated the reasons for the difference in stability between the discrete equations [6].

In (partial) differential equations, the stability and convergence of numerical solutions are necessary for the correctness of the solutions. Although we have proposed highly accurate numerical solutions for the semilinear Klein–Gordon equation [4, 5, 6], we have not quantitatively evaluated the stability and convergence of the solutions. In this paper, we propose some quantitative evaluation methods for the stability and convergence of the solutions for the semilinear Klein–Gordon equation in the flat spacetime.

Indices such as $(i,j,\dots)$ run from 1 to $n$, where $n$ is the spatial dimension. We use the Einstein convention of summation of repeated up–down indices in this paper.

2 Semilinear Klein–Gordon equation

The semilinear Klein–Gordon equation with the power-law nonlinear term in the flat spacetime is

[TABLE]

where $\phi$ is the dynamical variable, $\delta^{ij}$ is the Kronecker delta, $m$ is the mass, $c$ is the speed of light, $\hbar$ is the Dirac constant, $p$ is an integer larger than 2, and $\lambda$ is a constant and has a physical dimension of $1/(\text{length})^{2}$. When performing numerical calculations, the canonical form is preferable since it is a system of first-order equations in time. Moreover, it is easy to confirm the accuracy of the numerical calculations since there is a constraint with respect to time such as the total Hamiltonian. The Hamiltonian density of (1) is given by

[TABLE]

where $L_{0}$ is a constant value that makes the physical dimension of $\mathcal{H}$ into an energy dimension and $\psi$ is the canonical momentum of $\phi$. Then, the canonical equations of (1) are

[TABLE]

The discretized equations of (2), (3), and (4) can be respectively defined as

[TABLE]

[TABLE]

where (ℓ) means the time index, (k) means the space index, and $\boldsymbol{k}=(k_{1},\dots,k_{n})$. $\widehat{\delta}^{\langle 1\rangle}_{i}$ is the first-order central difference operator defined as

[TABLE]

Note that (5)–(7) are called Form I in [6]. Here, $\Delta x^{i}$ is the $i$ th grid range. If $n=3$, for example, $\Delta x^{1}=\Delta x$, $\Delta x^{2}=\Delta y$, and $\Delta x^{3}=\Delta z$. The nonlinear term can be expressed as

[TABLE]

The total Hamiltonian $\int_{\mathbb{R}^{n}}\mathcal{H}\,dx^{n}$ at a discrete level is preserved using (6) and (7) [6].

3 Quantitative evaluations of stability and convergence

In this paper, the word “stable simulation” means that no vibration occurs in the waveform of $\phi$. Moreover, to quantitatively evaluate stability, we define

[TABLE]

and sum of $|d\phi^{(\ell)}_{(\boldsymbol{k})}|/(\text{grid})$ if $d\phi^{(\ell)}_{(\boldsymbol{k})}$ satisfies the condition

[TABLE]

over $\boldsymbol{k}$, where $\hat{s}^{+}_{i}$ is the discrete operator that shifts the space forward. We call this value $SV_{g}$, which is determined for each grid, and consider the simulation stable when $SV_{g}$ is less than or equal to the threshold $\varepsilon_{\mathrm{s}}$. $SV_{g}$ represents the number of vibrations per grid weighted by $|d\phi^{(\ell)}_{(\boldsymbol{k})}|$. We study the appropriate value of $\varepsilon_{\mathrm{s}}$ in Section 4.

The word “convergence” means that $\phi$ approaches the exact solution with an increasing number of grids. To quantitatively determine convergence, we define the relative errors of $\phi$:

[TABLE]

where $\phi_{g}(x)$ is the value of $\phi$ for each grid number and $\phi_{\mathrm{G}}(x)$ is that for the maximum grid number. Since (6) and (7) have the second-order convergence with respect to the number of grids [6], we define the difference in $CV_{g}(t)$ from the second-order convergence as

[TABLE]

where $\bar{\mathrm{G}}$ is the second largest grid number. If $DCV_{g}(t)$ is less than or equal to the threshold $\varepsilon_{\mathrm{c}}$, we decide that the convergence of the simulation is satisfied. We also study the appropriate value of $\varepsilon_{\mathrm{c}}$ in Section 4.

4 Numerical results

In this section, we perform some simulations using the settings given below. The initial conditions are set as $\phi(x,\bar{\boldsymbol{x}})=\phi(x)=A\cos(2\pi x)$ and $\psi(x,\bar{\boldsymbol{x}})=\psi(x)=2\pi A\sin(2\pi x)$ for $x\in\mathbb{R}$ and $\bar{\boldsymbol{x}}\in\mathbb{R}^{n-1}$ with $A=2,3$ and $-1/2\leq x\leq 1/2$. The boundary is periodic. The physical parameters are $c=\hbar=L_{0}=1$. The spatial dimension is $n=3$. The grid ranges are $(\Delta x,\Delta t)=(1/250,1/2500)$, $(1/500,1/5000)$, $(1/1000,1/10000)$, $(1/2000,1/20000)$, $(1/4000,1/40000)$, and $(1/8000,1/80000)$. The simulation time is $0\leq t\leq 1000$. The number of exponents of the nonlinear term is $p=5$ and the coefficient parameter is $\lambda=1$. The mass $m$ ranges from $3.9$ to $4.2$ when $A=2$ and from $7.6$ to $8.2$ when $A=3$.

Fig. 1 shows $\phi$ with $A=2$ and $m=3.9$ to $4.2$. The vibration seems to occur at $t\geq 500$ for $m=4.0$ and at $t\geq 700$ for $m=4.1$. On the other hand, no vibration appears to occur for $m=3.9$ and $4.2$.

Fig. 2 shows $\phi$ with $A=3$ and $m=7.6$ to $8.2$. The vibration seems to occur at $t\geq 900$ for $m=7.8$, at $t\geq 300$ for $m=7.9$, and at $t\geq 400$ for $m=8.0$. On the other hand, no vibrations appear to occur for $m=7.6$, $7.7$, $8.1$, and $8.2$.

To quantitatively evaluate stability, we investigate the appropriate value of $\varepsilon_{\mathrm{s}}$. The value of $\varepsilon_{\mathrm{s}}$ is varied to $0.001$, $0.06$, $0.12$, $0.18$, $0.24$, and $0.30$. We summarize the results of the stability in Table 1.

$\bigcirc$ means that $SV_{8000}\leq\varepsilon_{\mathrm{s}}$ is always satisfied at $0\leq t\leq 1000$. On the other hand, the values represent the times when $SV_{8000}>\varepsilon_{\mathrm{s}}$.

Fig. 3 shows the convergence of $\phi$ for $A=2$ and $m=3.9$ to $4.2$. The convergence seems not satisfied for either $m=4$ or $m=4.1$.

On the other hand, Fig. 4 shows the convergence for $A=3$ and $m=7.6$ to $8.2$. The convergence seems not satisfied for $m=7.7$ to $8.1$.

To quantitatively evaluate convergence, we calculate $DCV_{g}(t)$ using (12) at various $\varepsilon_{\mathrm{c}}$ values from $0.1$ to $0.4$. Then, $\phi_{\mathrm{G}}(x)=\phi_{8000}(x)$ and $CV_{\bar{\mathrm{G}}}(t)=CV_{4000}(t)$ since the maximum grid number is $8000$ and the second largest grid number is $4000$ in these simulations. We summarize the results of the convergence in Table 2.

$\bigcirc$ means that $DCV_{2000}\leq\varepsilon_{\mathrm{c}}$ is always satisfied at $0\leq t\leq 1000$. On the other hand, the values represent the times when $DCV_{2000}>\varepsilon_{\mathrm{c}}$.

5 Conclusion and discussion

We showed some simulations of the semilinear Klein–Gordon equation with the power-law nonlinear term in the flat spacetime. The simulations were performed using the discrete equation, which was constructed by a structure-preserving scheme for various mass $m$ values from $3.9$ to $4.2$, where the amplitude of the initial value was $A=2$ and $m$ ranged from $7.6$ to $8.2$ when $A=3$. We proposed quantitative evaluation methods for stability and convergence.

For the threshold of stability, $\varepsilon_{\mathrm{s}}$, an appropriate value must be selected so that the time when vibration occurs in Figs. 1-2 and the time when $SV_{8000}>\varepsilon_{\mathrm{s}}$ are the same, for each amplitude $A$. In the case of $A=2$, from Fig. 1, the vibration appears to occur at $t\geq 500$ for $m=4.0$ and at $t\geq 700$ for $m=4.1$. Therefore, $\varepsilon_{\mathrm{s}}$ is adopted $0.24$ for $A=2$. In the case of $A=3$, from Fig. 2, the vibration appears to occur at $t\geq 900$ for $m=7.8$, at $t\geq 300$ for $m=7.9$, and at $t\geq 400$ for $m=8.0$. Therefore, $\varepsilon_{\mathrm{s}}$ is also adopted $0.24$ for $A=3$. On the other hand, for the threshold of convergence, $\varepsilon_{\mathrm{c}}$, there are differences from $0.1$ to $0.4$ in Table 2. $\varepsilon_{\mathrm{c}}$ is a value for quantitatively judging the second-order convergence of the solutions. For $A=2$ and $3$, there is no significant change in the number of times at which $\varepsilon_{\mathrm{c}}\geq 0.15$ and $\varepsilon_{\mathrm{c}}\geq 0.3$, respectively. Therefore, we adopt $0.15$ and $0.3$ as the thresholds for $A=2$ and $3$, respectively. Regarding $\varepsilon_{\mathrm{c}}$ for $A=3$ being greater than that for $A=2$, this means that the convergence becomes worse as the amplitude of the initial value increases. It seems that the large initial amplitude due to the effect of nonlinear terms worsens the convergence of the solutions. Note that the appropriate values of $\varepsilon_{\mathrm{s}}$ and $\varepsilon_{\mathrm{c}}$ depend on the parameters of the numerical calculation, such as the amplitude of the initial value and the mass. Thus, we have to investigate the appropriate values of thresholds under different numerical calculation conditions.

In this study, we only investigated in the flat spacetime. What we would like to investigate next in our future work is in a curved spacetime. The range of values for $m$ and $A$ is limited in this paper. Future work will involve investigating the stability and convergence over a wider range of values for $m$ and $A$.

Acknowledgments

T.T. and M.N. were partially supported by JSPS KAKENHI Grant Numbers 21K03354 and 24K06855. T.T. was partially supported by JSPS KAKENHI Grant Number 24K06856.