Structure-preserving numerical calculation of wave equation for a vector field

TL;DR

The paper tackles the challenge of stable, constraint-preserving numerical integration for the Proca equation, a vector-field wave equation derived from a Stueckelberg action. It introduces a structure-preserving discretization (SPS) that enforces the primary constraint $\mathcal{C}_1$ and the Gauss-like constraint $\mathcal{C}_2$ at the discrete level, and shows the discrete total Hamiltonian $H_C^{(\ell)}$ is conserved under suitable boundary conditions. A comparison with a standard discretization (SS) demonstrates that SPS yields smaller constraint violations, better energy conservation, and more stable long-time behavior in 3D tests. The work suggests that SPS is a robust approach for high-precision simulations of constrained vector-field dynamics and points toward extensions to nonlinear cases and deeper stability analyses.

Abstract

For the Proca equation, which is a wave equation for a vector field, we derive the canonical formulation including constraints from the Stueckelberg action and propose discrete equations with a structure-preserving scheme for conserving the constraints at the discrete level. Numerical simulations are performed using these discrete equations and other discrete equations with a standard scheme. We show the results obtained using the structure-preserving scheme and provide more accurate and stable numerical solutions.

Figures (4)

• Figure 1: $A^1$ with $\Delta x=\Delta y = 1/200$ at $t=19$, $20$, and $21$. The top panels are obtained with SS and the bottom ones with SPS. The left panels are at $t=19$, the middle ones at $t=20$, and the right ones at $t=21$.
• Figure 2: $A^1$ with $\Delta x=\Delta y=1/200$ on $x=y$ plane. The left panel is for SS and the right one is for SPS. For SS, the sizes of $A^1$ range approximately from $-0.8$ to $0.8$, whereas for SPS, they range approximately from $-1$ to $1$. The vibration seems to occur in the waveform obtained with SS.
• Figure 3: L2 norm of $\mathcal{C}_1$ and $\mathcal{C}_2$. The horizontal axis indicates time and the vertical one is the $\log_{10}$ of the L2 norm value. The left panel is for $\mathcal{C}_1$ and the right one is for $\mathcal{C}_2$. The dotted line is for SPS and $\Delta x=1/50$, the solid line is for SPS and $\Delta x=1/100$, the dashed line is for SPS and $\Delta x=1/200$, the dashed dotted line is for SS and $\Delta x=1/50$, the dashed double-dotted line is for SS and $\Delta x=1/100$, and the dashed triple-dotted line is for SS and $\Delta x=1/200$.
• Figure 4: Relative errors of $H_C$ against initial value. The vertical axis is $\log_{10}|(H_C-H_C(0))/H_C(0)|$. The others are the same as those in Fig. \ref{['fig:Const1-2']}.