Time-Dilation Methods for Extreme Multiscale Timestepping Problems

TL;DR

This work introduces a continuous space-time dilation framework that multiplies the local time derivative by a factor $a({\bf x},t)$ to slow dynamics in selected regions, allowing substantially larger global timesteps without distorting local steady-state behavior. By reformulating evolution equations into a dilated conservative form and providing explicit criteria, flux/source handling, and self-correction schemes, the method unifies and generalizes reduced-speed-of-light, slow-down techniques, and cyclic zoom approaches. Validation on idealized Bondi and Evrard collapse tests, plus a challenging multi-physics AGN scenario, demonstrates correct steady-state behavior and orders-of-magnitude speedups (up to $\sim 10^4$–$10^6$ in some cases) while highlighting limitations and necessary calibration. The approach offers a flexible, easily-implementable tool for tackling extreme temporal multi-scale problems in astrophysics, with potential impact on simulations ranging from BH accretion to cosmological mega-structures, albeit as a complement rather than a complete replacement for full-fidelity simulations.

Abstract

Many astrophysical simulations involve extreme dynamic range of timescales around 'special points' in the domain (e.g. black holes, stars, planets, disks, galaxies, shocks, mixing interfaces), where processes on small scales couple strongly to those on large scales. Adaptive resolution, multi-physics, and hybrid numerical methods have enabled tremendous progress on the spatial, physics, and numerical challenges involved. But often the limiter for following the long timescales of global evolution is the extremely short numerical timestep required in some subdomains (which leads to their dominating simulation costs). Recently several approaches have been developed for tackling this in problems where the short timescale solution is sampled and then projected as an effective subgrid model over longer timescales (e.g. 'zooming in and out'). We generalize these to a family of models where time evolution is modulated by a variable but continuous in space-and-time dilation/stretch factor $a({\bf x},\,t)$. This extends previous well-studied approaches (including reduced-speed-of-light and binary orbital dynamics methods), and ensures that the system comes to correct local steady-state solutions, and derive criteria that the dilation factor/timesteps/resolution must obey to ensure good behavior. We present a variety of generalizations to different physics or coupling scales. Compared to previous approaches, this method makes it possible to avoid imprinting arbitrary scales where there is no clear scale-separation, and couples well to Lagrangian or Eulerian methods. It is flexible and easily-implemented and we demonstrate its validity (and limitations) in test problems. We discuss the relationship between these methods and physical time dilation in GRMHD. We demonstrate how this can be used to obtain effective speedup factors exceeding $\gtrsim 10^{4}$ in multiphysics simulations.

Figures (4)

• Figure 1: Heuristic illustration of the method proposed here (§ \ref{['sec:timeline']}). Left: Standard evolution. Variables ${\bf U}_{i}$ are updated according to some time update $\mathcal{F}_{i}$, with a typical CFL-limited $\Delta t_{i} = [\Delta t_{i}]_{0}$, which here is small in some "inner" zone and large in some "outer" zone. We illustrate how cells in those zones would be advanced along the global timeline in parallel. Right: Evolving the dilated Eq. \ref{['eqn:method']}. All fluxes/updates are "slowed down" by $a = a({\bf x},\,t)$, allowing for a larger CFL-limited timestep $\Delta t_{i} = a_{i}^{-1} [\Delta t_{i}]_{0}$. Equivalently, on a global timeline update of $\Delta t_{i} = a_{i}^{-1} [\Delta t_{i}]_{0}$, cell $i$ experiences an effective timestep $a_{i} \Delta t_{i}$ in its frame, which is used to calculate the update to ${\bf U}_{i}$, while the cell is moved along the global timeline by $\Delta t_{i}$ (effectively "stretching" each timestep by a factor $a_{i}^{-1}$ on the timeline). The outer boundary cells have $a = 1$ so experience no dilation. Inner cells still take smaller updates on the global timeline (obeying conditions in § \ref{['sec:criteria']}), compared to outer cells, so they can respond to secular evolution or continuously changing information.
• Figure 2: 3D secularly-evolving spherical (Bondi) accretion test (§ \ref{['sec:demo.test:bondi']}). We initialize a uniform-density finite-mass periodic box of isothermal gas in a Keplerian, accreting potential, and allow it to evolve. We plot density and inflow velocity profiles at a time where a significant fraction of the box mass has been depleted. The time-dilation methods ($a < 1$) reproduce the solutions of the standard (no dilation, $a=1$) method up to integration errors. They capture the correct Bondi-like steady-state but also the continuous secular evolution as the box mass supply (hence $\rho_{\infty}$) depletes slowly.
• Figure 3: 3D non-equilibrium collapse/shock/hydrostatic equilibrium (Evrard) test (§ \ref{['sec:demo.test:evrard']}). We initialize a spherical self-gravitating cold adiabatic cloud which collapses, shocks, bounces back, and oscillates before slowly reaching hydrostatic equilibrium. Top: Early time where all radii are near-maximally non-steady-state. A strong shock has formed but incompletely propagated through. The reference solution has $a=1$. Dilation methods produce weakly different evolution at small radii where $a<1$. However, these simply reflect the "slowed" evolution of the system, reflecting the exact solution at slightly earlier times (shown). The time lag can be smaller than $a$ (e.g. just $\sim 20\%$ in time here at $r \sim 10^{-3}$, where $a \sim 0.03$), owing to its dependence on $r$. We compare a run which forces $\tilde{\bf S}\rightarrow 0$ in Eq. \ref{['eqn:hyper']}, i.e. places $a$ inside $\nabla \cdot {\bf F}$ instead of with $D_{t} a$: this restores the traditional definitions of conserved quantities but produces qualitatively incorrect evolution. Middle: Later time where the system is relaxing to equilibrium. The dilated and un-dilated methods relax to the same state. Bottom: Time evolution of mean density inside $r<0.3$. Runs dilated on smaller scales agree with exact solutions, other runs are slowed down as expected before reaching equilibrium.
• Figure 4: Re-runs for a duration $\sim 10^{4}\,G M_{\rm BH}/c^{3}$ of a multi-physics, multi-scale simulation of quasar accretion (§ \ref{['sec:demo:agn']}). Top: Radial profiles (in $r_{g} \equiv 2\, G M_{\rm BH}/c^{2}$) of different quantities ($90\%$ range shaded; median lines), in the original simulation (dashed) and a re-run with time dilation (solid). The time is chosen so there is a steady-state disk down to the ISCO. Middle: Same but chosen at a different time when the system goes into a strong radiation-pressure driven outburst with a MAD-like inner cavity. Bottom: CPU cost to completion of the runs tested with different $a(r)$, in terms of an "effective" $a_{\rm eff}$ at $\sim 10\,$times the inner boundary (see text for details). The time-dilated simulations appear to behave similarly to their un-dilated counterparts, but are orders-of-magnitude less expensive.