Numerical Schemes

The Numerical section controls the spatial discretization of convective and viscous fluxes and provides parameters to stabilize high-speed or shock-dominated flows.

Overview

This block specifies:

• conv_scheme – discretization of the convective terms
• conv_scheme_ibm – convective scheme used at immersed boundary (IBM) surfaces
• visc_scheme – viscous flux discretization
• Stabilization – tuning parameters for hybrid convective schemes

Convective Scheme (conv_scheme)

Convective schemes fall into two broad categories:

• Ordinary schemes – a single discretization (e.g. pure WENO or MUSCL2)
• Hybrid schemes – fourth-order central differencing in smooth regions, blended automatically with upwind WENO fluxes near shocks or turbulence using a dilatation-based shock sensor

For ideal_gas the available schemes are:

Scheme	Type	Notes / Recommended use
central4_charweno	Hybrid	For turbulence and strong shocks. Fourth-order central base with characteristic WENO3 upwinding. Recommended option for supersonic LES.
central4_charweno_lowdiss	Hybrid	As above but with a lower-dissipation, non-conservative flux. For LES or very low Mach numbers.
central4_weno3fvs	Hybrid	Uses WENO3 with flux-vector splitting; robust and stable, slightly higher dissipation than FDS. Recommended option for subsonic LES.
central4_weno3fds	Hybrid	Uses WENO3 with flux-difference splitting; slightly lower dissipation, slightly less robust.
central4_weno3fds_hllc	Hybrid	Variant using an HLLC flux; niche option for extreme shocks.
central4_weno3fvs_lowdiss	Hybrid	Non-conservative, lower-dissipation FVS variant.
central4_weno3fds_lowdiss	Hybrid	Non-conservative, lower-dissipation FDS variant.
central4	Ordinary	Pure fourth-order central, kinetic/entropy preserving. Mostly for research.
weno3	Ordinary	Pure 3rd-order WENO (no central blending). Robust and moderately dissipative.
charweno	Ordinary	Pure 3rd-order characteristic WENO; handles strong shocks, higher dissipation. Recommended option for inviscid hypersonic simulations.
charweno_ausm	Ordinary	Characteristic WENO with AUSM flux. Good for weak shocks and jets, less robust on strong shocks.
muscl2	Ordinary	Classical 2nd-order MUSCL. Legacy or low-cost runs only.
first_order	Ordinary	Very diffusive; emergency-only for debugging or pathological cases.
none	Ordinary	Disables convection entirely. For manufactured-solution or special test cases only.

Other physical models (e.g. air_CO2, air_SF6, air5, air11) may support different sets of convective schemes.
These will be documented in future updates.

Convective Scheme near IBM (conv_scheme_ibm)

This controls the discretization near immersed boundaries:

• auto (default): mimics the interior scheme while maintaining stability with cut-cell geometry.
• first_order: forces first-order upwind near IBM boundaries for maximum robustness (more diffusive).

Viscous Scheme (visc_scheme)

Options:

• visc2 (recommended): standard second-order viscous discretization.
• none: disables viscous fluxes, suitable only for inviscid (Euler) test problems.

Stabilization

Hybrid convective schemes introduce additional stabilization controls:

Parameter	Meaning	Guidance
ducr_shield	Physical distance outside which the hybrid sensor is set to 1.0 (fully upwind).	Normally leave unset.
ducr_multiplier	Scales the dilatation-based shock sensor.	Usually 1.0.
ducr_cutoff	If the dilatation sensor is below this value, it is set to zero.	Lowering increases dissipation.
minimum_value	Minimum blending value anywhere in the flow.	Low Mach: ~1e-3–5e-3. Hypersonic: 0.01–0.1. Higher = more stable.
minimum_value_ibm	Minimum blending value applied at immersed boundaries.	Set similar to minimum_value.

These parameters allow tuning for challenging cases such as very low Mach numbers or hypersonic flows with strong shocks.

Reflux

reflux (default: false)
Enables conservative flux correction across coarse–fine grid interfaces (AMR).
When true, the solver ensures that net fluxes are exactly conserved between adjacent blocks of different resolution.
This slightly increases computational cost and memory but improves global conservation of mass, momentum, and energy, especially for long-time or high-accuracy runs.

Prototype

Refluxing on multi-GPU simulations is currently not supported. It's not recommended to use refluxing.

Example

Numerical
{
    conv_scheme     = "central4_charweno"
    conv_scheme_ibm = "auto"
    visc_scheme     = "visc2"
    reflux          = false

    Stabilization
    {
        ducr_shield       = 1.0
        ducr_multiplier   = 1.0
        ducr_cutoff       = 0.25
        minimum_value     = 0.01
        minimum_value_ibm = 0.01
    }
}

This configuration

• uses a fourth-order central + characteristic WENO hybrid scheme for the convective fluxes,
• sets the immersed-boundary scheme to automatically match the interior scheme, and
• applies second-order viscous discretization.