Hybrid LAD-GBR Solver for High-Speed Flows

• The paper introduces a hybrid solver that dynamically switches between LAD and GBR to optimize stability and efficiency in high-speed flow simulations.
• It leverages local artificial diffusivity for stabilization near discontinuities while using high-order gradient reconstruction in smooth regions.
• Benchmark results reveal a 1.5–1.7× speed-up with reliable shock and turbulence capture, making the method cost-effective for complex CFD problems.

A hybrid LAD-GBR solver is a numerical algorithm designed for the robust and efficient simulation of high-speed compressible flows, especially in regimes involving strong shock-boundary layer interactions. It blends local artificial diffusivity (LAD) with centralised gradient-based reconstruction (C-GBR) techniques, switching between them dynamically to exploit the favorable cost, robustness, and accuracy tradeoffs of each method (Kumar et al., 22 Nov 2025).

1. Fundamental Formulation

The hybrid LAD-GBR solver is built on the compressible Navier–Stokes or Euler equations in conservative form, with augmentation by artificial diffusion terms when required for stabilization. The semi-discrete governing equations in a generic Cartesian or mapped curvilinear system are: $$\frac{\partial U}{\partial t} + \nabla\cdot F^c(U) = \nabla\cdot\left[ F^v(U) + D^{\mathrm{LAD}}(U) \right]$$ where:

• $U = [\rho, \rho u, \rho v, \rho w, E]^T$ is the conservative state,
• $F^c$ and $F^v$ are convective and physical viscous fluxes,
• $D^{\mathrm{LAD}}$ represents LAD-generated artificial viscosities ($\mu^*, \beta^*, \kappa^*$), inserted only near discontinuities.

Time integration is executed using explicit TVD Runge–Kutta schemes of third or fourth order. Spatial derivatives are evaluated using either central finite differences (LAD) or characteristic-based high-order stencils (GBR), depending on the activated mode (Kumar et al., 22 Nov 2025).

2. Constituent Numerical Methods: LAD and C-GBR

Local Artificial Diffusivity (LAD)

In LAD, each physical transport coefficient is supplemented by an artificial term: $$\mu = \mu_{\mathrm{phys}} + \mu^*, \quad \beta = \beta_{\mathrm{phys}} + \beta^*, \quad \kappa = \kappa_{\mathrm{phys}} + \kappa^*$$ with \begin{align*} \mu^* &= C_\mu\,\overline{\rho \left| \sum_{l=1}^3 \frac{\partial^4 S}{\partial\xi_l^4}\, \Delta\xi_l^4\,\Delta_{l,\mu}^2 \right|} \ \kappa^* &= C_\kappa\,\overline{ \frac{\rho\,c_s}{T} \left| \sum_{l=1}^3 \frac{\partial^4 e}{\partial\xi_l^4}\, \Delta\xi_l^4\,\Delta_{l,\kappa} \right|} \ \beta^* &= C_\beta\,\overline{ \rho\,f_{sw} \left| \sum_{l=1}^3 \frac{\partial^4 (\nabla\cdot u)}{\partial\xi_l^4}\, \Delta\xi_l^4\,\Delta_{l,\beta}^2 \right| } \end{align*} where $f_{sw}$ is a Ducros-type sensor activating bulk viscosity in zones of compression. Constants are typically $C_\mu = 0.002$, $C_\beta = 1.75$, and $C_\kappa = 0.01$. The overbar denotes a truncated-Gaussian smoothing filter.

Gradient-Based Reconstruction (C-GBR)

C-GBR computes high-order pointwise gradients of primitive variables and applies a wave-appropriate discontinuity sensor: $$\phi = \frac{ (\nabla\cdot u)^2 }{ (\nabla\cdot u)^2 + (\nabla\times u)^2 + \epsilon }$$

• For acoustic modes, if $\overline{\phi} > 0.01$, a monotonicity-preserving (MP) nonlinear stencil is enforced, else a linear stencil is used.
• Shear/vortical and entropy/contact waves are handled via similar or monotonicity-based criteria.
• Interface states are constructed by blending linear and nonlinear reconstructions, with centralization except for upwinded acoustic waves.
• Inviscid fluxes are evaluated with the HLLC approximate Riemann solver. Wave speeds and star-region projections follow the classical HLLC methodology.

3. Hybrid Blending Algorithm

The hybrid strategy switches between LAD and GBR by monitoring local flow quantities for under-resolved or troubled regions: for each time step: for each RK stage: if ρ_min < 0.5·ρ_ref (or p_min < 0.5·p_∞): use GBR solver else: use LAD solver Here, $\rho_{\mathrm{ref}}$ is problem-specific (e.g., minimum initial quadrant density for Riemann problems or the freestream density for shock/boundary-layer flows). The default execution uses LAD, with a global switch to GBR whenever sub-threshold density or pressure is detected, then reverting to LAD as conditions stabilize. This global trigger is designed for algorithmic simplicity, though plausible future refinements suggest fully localizable triggers (Kumar et al., 22 Nov 2025).

4. Stability Properties and Performance

Pure LAD solvers diverge under strong discontinuities:

• Le Blanc shock tube ($\rho$ jump ∼ $10^3$, $p$ jump ∼ $10^9$),
• Strong 2D Riemann (max $p$ ratio ∼ 51),
• Compressible triple-point,
• LES of $M=2.9, \theta=24^\circ$ and $M=7.2, \theta=8^\circ$ compression ramps.

C-GBR solvers remain stable and accurately capture both shocks and turbulent structures without spurious oscillations.

The hybrid solver inherits LAD's cost advantage in smooth regions and GBR's robustness only where instability is anticipated. In practical simulations, such as the $M=2.9, \theta=24^\circ$ ramp test, the hybrid LAD-GBR solver executed GBR for only 17.6% of the simulation runtime, yielding a 1.67× net speed-up relative to pure GBR with indistinguishable shock, wall-pressure, and skin-friction predictions (Kumar et al., 22 Nov 2025).

5. Quantitative Benchmarks

The following summarizes the principal runtime and accuracy benchmarks:

Scheme	Relative Speed (vs MEG6)	Stability	Accuracy in STBLI
LAD-C6/E4	1.17–2.3× faster	Diverges on strong shocks	Comparable (when stable)
GBR-MEG6/MIG4	Baseline	Always stable	Reference (gold standard)
Hybrid LAD-GBR	1.5–1.7× net speed-up	Always stable	Matches GBR results

On NVIDIA A100 GPUs, explicit LAD (LAD-E4) achieves 1.9–2.0× the speed of MEG6 GBR and 2.2–2.3× over MIG4 GBR. The hybrid's speed-up increases with the proportion of stable, smooth regions in the domain (Kumar et al., 22 Nov 2025).

6. Applicability and Impact in Shock-Turbulence Simulations

The hybrid LAD-GBR solver has demonstrated efficacy in:

• Compressible triple-point and strong Riemann problems, recovering correct flow structures when LAD would otherwise fail.
• Large-eddy simulations (LES) of STBLI in supersonic and hypersonic flows, delivering stable, high-fidelity wall-pressure and skin-friction predictions, with reattachment points within ±2% of experimental and DNS references.
• Scenarios where pure LAD would require significant parameter tuning or fail entirely, while GBR's cost is prohibitive for large domains.

This suggests the method lowers the cost barrier for accurate simulation of complex shock-turbulence interactions without sacrificing robustness or fidelity, especially in parameter regimes (strong shocks, contact discontinuities) that traditionally required expensive, global nonlinear reconstructions (Kumar et al., 22 Nov 2025).

7. Limitations and Prospective Developments

Pure LAD retains limitations in the presence of steep discontinuities, necessitating dynamic switching. The current hybrid implementation deploys a global mode switch—the structure of which is amenable to further localization for even greater computational efficiency.

A plausible implication is that the hybrid LAD-GBR design paradigm can be generalised to other classes of high-speed CFD, and to architectures where cost trade-off is paramount, such as GPU-accelerated LES. Ongoing work may extend the approach to localized switching at the cell or block level, further reducing GBR's computational footprint (Kumar et al., 22 Nov 2025).

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References:

• "Assessment of Gradient-based Reconstruction and Artificial Diffusivity Methods in Simulating High-Speed Compressible Flows" (Kumar et al., 22 Nov 2025)