Time integration as filtering: a space-time discretization-aware LES formulation

Technical Summary: Time Integration as Filtering in Space–Time Discretization-Aware LES

Problem Statement
In Implicit Large Eddy Simulation (ILES), the spatial filter is often identified with the computational grid itself. Traditional "discretization-aware" LES formulations address the turbulence closure problem by deriving an exact residual stress based on the "filter-swap" property: the coarse central finite difference is recognized as a top-hat-filtered exact spatial derivative. However, existing formulations treat time integration as a separate, post-processing step. Consequently, the time-discretization error lies outside the exact residual-stress guarantee. As time steps increase (higher CFL numbers), standard space-only closures degrade because they fail to account for the specific error introduced by the time integrator, particularly when using explicit schemes like Forward Euler.

Methodology
The paper extends the spatial filter-swap concept to space–time by observing that the Forward-Euler time difference is itself a (one-sided, top-hat) filtered exact time derivative.

1. Space–Time Filter-Swap: By applying a space–time filter $(\cdot)_{h,\tau} = (\cdot)_\tau \circ (\cdot)_h$ to the scalar conservation law, the authors derive an exact discrete law where both spatial and temporal derivatives are replaced by their filtered counterparts.
2. Flux Decomposition: The exact space–time flux is decomposed into the numerical flux plus a residual flux, $r_{h,\tau}$. This residual is split into four distinct error terms:
   • $r^{LES}_h$: The classical LES commutator (grid filtering vs. nonlinearity).
   • $r^{num}_h$: The numerical flux error.
   • $r^{div}_h$: A correction for discrete divergence.
   • $r^{time}_\tau$: A new term representing the flux time-quadrature error (the difference between the step-averaged flux and the instantaneous flux).
3. Modeling: The authors propose a space–time discretization-aware eddy-viscosity model. They expand the time-quadrature error term, $r^{time}_\tau$, showing it behaves as a diffusive flux proportional to $\tau (f'(u))^2 \partial_x u$. This is identified as a Lax–Wendroff-type diffusion, which is non-negative by construction and scales linearly with the time step $\tau$. The final closure augments the standard Smagorinsky model with this temporal term.

Key Results
The framework was tested on viscous Burgers turbulence using a Forward-Euler finite-volume scheme with varying CFL numbers (0.06 to 0.89).

• A-priori Analysis: In the decomposition of the residual flux, the temporal term ($r^{time}_\tau$) grows significantly with the CFL number, becoming the largest single error term at practical CFL values (e.g., 0.59 of the residual at CFL 0.89 for central fluxes). Conversely, the spatial terms maintain fixed proportions regardless of the time step.
• A-posteriori Analysis:
  • Central Flux: Standard closures (no model, classic Smagorinsky, and space-only Smagorinsky) degrade rapidly as the CFL number increases, with errors rising from 0.1 to ~0.85. The proposed space–time-aware closure maintains a low, constant relative error (0.09) across the entire range of CFL numbers.
  • Upwind Flux: The upwind numerical flux is inherently over-dissipative. In this case, the temporal error (which is non-dissipative) partially cancels the over-dissipation of the upwind flux. While the space–time model remains accurate, it is slightly over-dissipative at large steps because it fits the cancellation effects independently rather than exploiting the fortuitous cancellation.

Significance and Claims
The paper claims that recognizing the Forward-Euler time difference as a top-hat filter allows for an exact discrete flux decomposition in space–time. The primary contribution is the identification of the flux time-quadrature error as a distinct, dominant term in the residual at coarse time steps.

• The authors demonstrate that a closure augmented with the leading-order term of this error (Lax–Wendroff diffusion) allows Forward-Euler simulations to remain accurate even at large time steps where space-only closures fail.
• The work establishes that the time integrator contributes exactly one new term to the residual (the quadrature error) and no separate temporal commutation term.
• The paper modestly notes that much of the time-quadrature error is non-dissipative and thus invisible to standard eddy-viscosity models, suggesting that machine-learned closures trained on the exact target or implicit/multi-stage integrators are natural future extensions.