Local Artificial Diffusivity (LAD)

• Local Artificial Diffusivity (LAD) is a numerical stabilization strategy that dynamically activates artificial viscosity and diffusivity based on high-order flow sensors to mitigate spurious oscillations.
• It augments physical transport coefficients in simulations of compressible, incompressible, and multiphase flows, ensuring minimal unphysical dissipation in well-resolved regions while stabilizing shocks and interfaces.
• LAD employs robust sensor designs and hybridization techniques, such as integrating gradient reconstruction methods, to balance computational efficiency with high-fidelity resolution across complex flow features.

Local Artificial Diffusivity (LAD) denotes a class of numerical stabilization strategies for high-fidelity simulations of compressible, incompressible, and multi-material flows, in which localized, solution-dependent artificial viscosity, diffusivity, or stress-diffusion terms are incorporated into the governing equations to control spurious oscillations near discontinuities, under-resolved gradients, or material interfaces. The distinguishing feature is the localization of these artificial transport coefficients—activation is governed by high-order, flow-aware sensors rather than imposed globally—thus minimizing unphysical dissipation in resolved regions while ensuring numerical stability across features such as shocks, contacts, vortices, and high-density interfacial layers.

1. Formulation and Fundamental Principles

Local artificial diffusivity schemes augment the baseline viscous, diffusive, or stress terms in the Navier–Stokes or related equations by adding artificial components that are activated locally in regions of strong gradients or non-smoothness. Formally, for a generic transport equation, the total viscosity, bulk viscosity, or diffusivity is written as a sum of the physical and artificial (LAD) contributions:

$$\mu = \mu_f + \mu^*,\quad \beta = \beta_f + \beta^*,\quad \kappa = \kappa_f + \kappa^*$$

where the artificial terms $\mu^*$, $\beta^*$, and $\kappa^*$ are computed from local, flow-sensitive sensors—typically involving high-order derivatives (e.g., fourth- or eighth-order, depending on context), filtered amplitudes, or tensor-norms of the relevant field, and are often further restricted by detector functions such as Heaviside gates, Ducros/strain-vorticity ratios, or locally normalized measures (Kumar et al., 22 Nov 2025, Jain et al., 2023, Dzanic et al., 2022, Ribeiro et al., 30 Apr 2025, Brill et al., 16 Mar 2025). This localization ensures that artificial dissipation is injected only where necessitated by unresolved sharp features, avoiding global laminarization or loss of fine-scale dynamics.

2. Sensors and Localization Strategies

LAD methods rely critically on the design of robust sensors to (i) distinguish between physical and numerical discontinuities, (ii) discriminate between shocks, contacts, vortical regions, and smooth areas, and (iii) control the magnitude and spatial extent of artificial coefficients. Common strategies include:

• High-Order Derivative Indicators: Artificial viscosities are set proportional to filtered norms of high-order derivatives (e.g., $\partial_\xi^4 S$ for strain, $\partial_\xi^4(\nabla\cdot u)$ for dilatation), with the amplitude providing a quantitative measure of under-resolution or Gibbs phenomena (Kumar et al., 22 Nov 2025, Ribeiro et al., 30 Apr 2025, Jain et al., 2023).
• Discontinuity Detectors: Contact and shock detection is achieved by sensors such as:
  • Ducros Sensor $$f_\beta = \theta^2 / (\theta^2 + a\,\omega^2 + \varepsilon)$$, with high values in compression-dominated (shock) regions, penalizing vorticity.
  • Contact Discontinuity Sensor $$f_D = |\nabla \rho|^2 / (|\nabla \rho|^2 + a(\theta^2 + \omega^2)(\rho/|\vec{u}|)^2 + \varepsilon)$$, designed to activate at contact surfaces while suppressing effects in shocks/vortices (Jain et al., 2023).
  • Q-Criterion/Low-Pressure Core Sensors: For accurate treatment of vortical structures in LES, auxiliary sensors based on the Q-criterion and a low-pressure threshold spatially localize artificial shear viscosity to expansion-driven vortex cores (Ribeiro et al., 30 Apr 2025).

Localization is mathematically achieved either by direct multiplicative gating of the raw artificial coefficient by the indicator (sensor) or by continuous normalization—e.g., in the "modified artificial-diffusivity" (MAD) scheme for polymeric flows, the local coefficient is scaled by the pointwise steepness indicator $$\chi(x, t) = \mathcal{Q}(x, t)/\mathcal{Q}_{\max}(t)$$, confining diffusion to regions of maximal stress gradient and seamlessly vanishing elsewhere (Dzanic et al., 2022).

3. LAD in Compressible and Multiphase Flows

In high-speed compressible flow solvers, LAD is used to inject grid-dependent artificial viscosity, bulk viscosity, and thermal diffusivity that target shocks, contacts, and under-resolved features:

• Artificial Bulk Viscosity ($\beta^*$) is localized to shocks or compression waves, with activation further limited by negative dilatation and vorticity-dilution sensors.
• Artificial Mass Diffusivity ($D^*$) is applied for contact discontinuities, with density-gradient-based sensors tailored to avoid unnecessary dissipation in regions governed by vorticity or shocks (Jain et al., 2023).

In multi-material and multi-phase contexts, LAD faces distinct challenges due to large density ratios, thermodynamic consistency, and the necessity to maintain immiscible interface thicknesses. Recent extensions introduce

• Bulk-Density Diffusion: A single artificial diffusivity acting on total $\rho Y_i$, preserving pressure–temperature equilibrium even under extreme density contrasts.
• Phase-Field Sharpening Terms: Artificial interface sharpening fluxes, derived from phase-field dynamics, maintain interfaces at grid-scale thickness and prevent excessive smearing (Brill et al., 16 Mar 2025).

These modifications demonstrate stable operation for Atwood numbers approaching unity, high-density ratio bubble advection, and Rayleigh–Taylor or shock–bubble interactions—regimes in which original mass-fraction-based LAD is prone to interface-misalignment or numerical breakdown.

4. Algorithmic Implementation and Numerical Performance

The canonical LAD workflow is structured for minimal computational overhead and compatibility with high-order schemes:

1. At each time step or Runge–Kutta substage, compute gradients and high-order derivatives of discriminative variables (e.g., $\nabla\rho$, $\partial^4S/\partial\xi^4$).
2. Filter the sensor fields as needed (Gaussian or compact-stencil filters).
3. Evaluate localization sensors and assemble artificial transport coefficients.
4. Add the resulting viscous, diffusive, or phase-field terms to the governing PDEs in a discrete, flux-consistent fashion—entropy preservation and kinetic-energy consistency are enforced with tailored flux pairs, such as the kinetic energy and entropy preserving (KEEP) central flux combined with localized Lax–Friedrichs type dissipative fluxes (Jain et al., 2023).
5. Advance the solution with explicit or semi-implicit time integrators, optionally applying high-order compact filters for suppressing residual high-wavenumber oscillations (Kumar et al., 22 Nov 2025, Brill et al., 16 Mar 2025).

Quantitative studies show that, for elastic turbulence and high Wi polymer flows, locally weighted LAD (or MAD) recovers the correct turbulent spectra and symmetry properties at a fraction of the resolution otherwise required to approach the vanishing-diffusivity limit (reducing grid requirements by factors of 10–20) and eliminating strong unphysical artifacts associated with globally applied artificial diffusion (Dzanic et al., 2022).

For compressible turbulent boundary layers at moderate Mach, LAD achieves mean flow and Reynolds stress profiles commensurate with more expensive gradient-based reconstruction (GBR) methods, and computational timing benchmarks confirm 1.17–2.3× speedups versus explicit/implicit C-GBR schemes. However, for shock-dominated, large separation flows (e.g., ramp compression at $M=2.9$ or $M=7.2$), pure LAD is prone to abrupt divergence due to unbounded density/pressure excursions, whereas hybrid LAD–GBR approaches recover robustness while retaining substantial computational advantage (Kumar et al., 22 Nov 2025).

5. Comparative Evaluation, Hybridization, and Limitations

Extensive comparative studies across benchmark cases and flow regimes yield the following summaries:

Method	Strengths	Weaknesses/Limitations
LAD	Simple, efficient, high-order-friendly	Divergence/failure in extreme shocks and large separations
C-GBR	Robust at all regimes	Computationally expensive, complex implementation
Hybrid (LAD+GBR)	Robust as C-GBR, fast as LAD in smooth regions	Field-wide switch only (no spatial blending); more complex data handling needed for local adaptation

• For flows dominated by moderate gradients, shocks, or interfaces, LAD is favored for its efficiency and plug-and-play nature in compact finite-difference and spectral codes.
• In vortex-dominated LES, the use of specialized sensors for shear viscosity suppresses excessive artificial momentum transport in boundary layers and wakes, permitting high-fidelity coherent structure resolution while maintaining the necessary stabilizing diffusion in expansion-driven cores (Ribeiro et al., 30 Apr 2025).
• For immiscible, multi-material, or extreme Atwood number flows, extended LAD schemes with interface sharpening and bulk-density diffusion enable stable, conservative simulation at grid-scale interface thickness, overcoming interface drift and thermodynamic inconsistency endemic to original mass-fraction-only approaches (Brill et al., 16 Mar 2025).

However, in the presence of extremely strong shocks, complex shock-vortex-contact interactions, or rarefaction-induced near-vacuum states, pure LAD may fail to suppress growth of high-wavenumber instabilities, necessitating recourse to more robust, albeit costlier, methods based on gradient reconstruction and Riemann solvers, or the use of dynamic hybrid field-wide or local switching (Kumar et al., 22 Nov 2025).

6. Applications and Implementation Guidelines

LAD methods have been applied across a spectrum of flow problems:

• Elastic turbulence simulations where high Weissenberg number flows require accurate resolution of steep stress gradients and preservation of underlying forcing symmetries (Dzanic et al., 2022).
• Compressible multi-material and multi-phase flows at high density ratios, where pressure–temperature equilibrium at interfaces and robust handling of species volume and mass fraction gradients are mandatory (Brill et al., 16 Mar 2025).
• LES/DNS of shock-boundary layer interactions, vortex-dominated wakes, and supersonic/hypersonic ramp flows, where judiciously tuned sensors enable accurate and stable computations with significant computational speedup over monotonicity and exact-Riemann based central gradient methods (Kumar et al., 22 Nov 2025, Ribeiro et al., 30 Apr 2025).

Essential implementation guidelines include:

• Consistent application of SPD-conserving schemes in viscoelastic and tensor PDE contexts.
• High-order spatial discretization, compatible with local filters and derivative computation for artificial coefficient assembly.
• Careful selection of sensor tuning constants, pressure/vorticity thresholds, and interface sharpening parameters—empirical studies indicate only mild sensitivity, allowing broad portability across solvers and grid types.
• Monitoring volume-integrated artificial dissipation for diagnostic assessment of over- or under-diffusion, especially as grid resolution and flow regime parameters are varied.

7. Ongoing Developments and Research Directions

Current research efforts aim to further improve LAD methodology along several axes:

• Hybridization: Refining hybrid LAD–GBR approaches by transitioning from field-wide switches to locally adaptive spatial blending, optimizing the balance between accuracy, robustness, and computational cost (Kumar et al., 22 Nov 2025).
• Unstructured Grids: Deployment of LAD on unstructured meshes, leveraging the lack of explicit field filtering or sensor smoothing requirements (Jain et al., 2023).
• Interface-Consistent Extensions: Generalization of LAD to N-material, compressible, and thermodynamically non-equilibrium systems using advanced consistency terms and phase-field interface sharpening (Brill et al., 16 Mar 2025).
• Sensor Development: Exploration of higher-order, machine-learning-guided, or dynamically adaptive sensors to distinguish physical from numerical non-smoothness across scales.

These developments position local artificial diffusivity as a central tool in the evolving landscape of high-fidelity, robust, and efficient computational fluid dynamics, while delineating clear regimes of applicability and known failure modes.