A hyperboloidal method for numerical simulations of multidimensional nonlinear wave equations: nonlinear tails

TL;DR

The paper develops and implements a hyperboloidal, conformally compactified framework to numerically study the nonlinear wave equation $\Box \Phi = \mu |\Phi|^{p-1}\Phi$ in $n+1$ dimensions without full spherical symmetry for $n=3$ and with SO$(n-1)$ symmetry in higher dimensions. It derives a regular first-order system for the rescaled field $\tilde{\Phi}$ and its conjugate $\tilde{\Pi}$ on hyperboloidal slices, establishing regularity at future null infinity $\mathrsfs{I}^+$ and an energy balance that accounts for energy flux through $\mathrsfs{I}^+$. Numerically, the authors employ a hybrid discretisation (radial fourth-order finite differences with angular pseudo-spectral methods), RK4 time stepping, Kreiss-Oliger dissipation, and Orszag 2/3 filtering to evolve the system up to $\mathrsfs{I}^+$. They compute tail decay exponents $q_{lm}$ across subcritical, critical, and supercritical regimes in $n=3$ and $n=5$ with SO$(n-1)$ symmetry, finding mode-dependent exponents that are independent of the azimuthal index in 3D and that agree with conjectured finite-radius and $\mathrsfs{I}^+$ scaling relations, thereby extending previous radial results to nonradial settings and highlighting the method's capacity to probe nonlinear tails in multi-dimensional NLWs.

Abstract

We consider the scalar wave equation with power nonlinearity in n+1 dimensions. Unlike most previous numerical studies, we go beyond the radial case and do not assume any symmetries for n=3, and we only impose an SO(n-1) symmetry in higher dimensions. Our method is based on a hyperboloidal foliation of Minkowski spacetime and conformal compactification. We focus on the late-time power-law decay (tails) of the solutions and compute decay exponents for different spherical harmonic modes, for subcritical, critical and supercritical, focusing and defocusing nonlinear wave equations.

Figures (10)

• Figure 1: Illustration of the hyperboloidal method. (a) Outgoing (solid blue) and ingoing (dotted red) characteristics $t=\pm r + \mathrm{const}$ of the wave equation. (b) The same characteristics plotted against the compactified coordinate $\tilde{r}$ defined in e:rtilde_intro. (c) Characteristics in terms of $\tilde{r}$ and the hyperboloidal time coordinate $\tilde{t}$ defined in e:ttilde_intro. (d) A few hyperboloidal slices $\tilde{t} = \mathrm{const}$ (evenly spaced in $\tilde{t}$), plotted in $(r,t)$ coordinates. The dots indicate a grid that is uniform in the compactified coordinate $\tilde{r}$. We have chosen $a=6$ in e:rtilde_intro and e:ttilde_intro, which corresponds to dimension $n=3$ and mean curvature constant $C=0.5$ in section \ref{['s:metric']}.
• Figure 2: Convergence test against an exact linear solution. (a) $n=3$ dimensions without symmetries, (b) $n=5$ with SO$(4)$ symmetry. Shown is the $L^2$ norm of the error of $\tilde{\Phi}$ w.r.t. the exact linear solution as a function of time for three different radial resolutions, from top to bottom: $N_{\tilde{r}}=250$ (blue), $N_{\tilde{r}}=500$ (red) and $N_{\tilde{r}}=1000$ (green).
• Figure 3: Energy balance for the nonlinear wave equation. (a) $n=3,\, p=5$, (b) $n=5,\, p=3$, both with SO$(n-1)$ symmetry. Shown is the energy $E(\tilde{t})$ (blue, monotonically decreasing), the integrated flux $-F(0,\tilde{t})$ (red, monotonically increasing) and their sum (green, nearly constant). Solid lines correspond to an evolution with a focusing nonlinearity ($\mu=-1$), dashed lines to a defocusing nonlinearity ($\mu=1$).
• Figure 4: Convergence test for the relative error in the energy balance of a defocusing $n=5$, $p=3$ NLW evolution. (a) Varying radial resolution $N_{\tilde{r}}=1000, 2000, 4000$ (from top to bottom) at fixed angular resolution $N_\theta=24$, (b) varying angular resolution $N_\theta=16, 20, 24$ (from top to bottom) at fixed $N_{\tilde{r}}=4000$.
• Figure 5: Ratio $E_\mathrm{pot}(\tilde{t})/E(\tilde{t})$ of the potential energy to the total energy as a function of time for the same evolutions as in figure \ref{['f:energybalance']}. (a) $n=3,\, p=5$, (b) $n=5,\, p=3$, both with SO$(n-1)$ symmetry. Solid lines correspond to an evolution with a focusing nonlinearity ($\mu=-1$), dashed lines to a defocusing nonlinearity ($\mu=1$).

Theorems & Definitions (1)

• Conjecture 1