numerical scheme

Default. One-dimensional: $\psi = \max\left(0, \dfrac{2ab + \varepsilon^2}{a^2 + b^2 + \varepsilon^2}\right)$. Smooth, symmetric, bounded in $[0, 1]$.

The recommended general purpose choice.

One-dimensional: $\psi = \max\left(0, \dfrac{4ab + \varepsilon^2}{(a + b)^2 + \varepsilon^2}\right)$. Slightly more compressive than van Albada.

One-dimensional: $\psi = \max\left(0, \min\left(1, \dfrac{4\min\left(\sigma a, \sigma b\right) + \varepsilon}{\left|a + b\right| + \varepsilon}\right)\right)$ with $\sigma = \mathrm{sign}(a + b)$. The least dissipative of the three.

$\psi = 1$. Unlimited reconstruction: second order everywhere, but not monotonicity preserving.

Appropriate only for smooth flows with no shocks or strong gradients.

Cell-based: $P(y) = \min(1, y)$. Barth & Jespersen (1989) as published. Bound preserving, but only $C^0$: the kink at $y = 1$ makes the limiter value flicker between cycles and can stall the residual. It has no smoothness deactivation, so it limits wherever the bound binds.

Cell-based: $P(y) = y - \tfrac{4}{27}y^3$ for $y < 3/2$, else 1. Michalak & Ollivier-Gooch. $C^1$, and $P(y) \leq \min(1,y)$, so never less monotonicity preserving than Barth-Jespersen. Deactivates in smooth flow via a grid-scaled threshold, so formal order survives refinement.

The recommended choice.

Cell-based: $P(y) = \min\left(1, (y^2+2y)/(y^2+y+2)\right)$. Smooth, but exceeds 1 for $y > 2$, so it needs the clip, which reintroduces a $C^0$ kink at $y = 2$.

Provided as a familiar cross-check.

Green-Gauss gradients from face values interpolated by inverse distance, or by a plain average when linear_gradients is set.

The default.

A least-squares fit to the differences between each cell and its face neighbours, weighted by face area.

The MUSCL $\kappa$-scheme with $\kappa = 1/3$, blending the limited cell-gradient extrapolation with the central difference across the face.

The default.

The unstructured MUSCL of Burg, $u_f = u_L + (1-\kappa)\,\mathbf{r}\cdot\nabla u_L + \tfrac{\kappa}{2}(u_R - u_L)$ with $\kappa = 1/3$, scaled by the limiter.

Accepted, but the solver currently treats it as muscl.

Roe's approximate Riemann solver. entropy_fix floors each wave speed at that fraction of the largest. Read by the compressible solver only.

The default.

The HLLC approximate Riemann solver, which restores the contact wave missing from HLL.

The Roe flux with its upwind dissipation scaled by a factor between roe_dissipation_sensor_minimum and 1, set per face by roe_low_dissipation_sensor.

Scale-resolving simulations, where full upwind dissipation would damp the resolved turbulence. With the sensor NONE it is the same as roe.

The HLLC flux with its dissipation scaled in the same way as roe low-diss.

The Rusanov (local Lax-Friedrichs) flux, which scales the dissipation by the largest local wave speed.

The most dissipative of the schemes.

Scales the dissipation by $1 - f_d$, where $f_d$ is the DDES shielding function of the turbulence model: full dissipation in RANS regions, the minimum in LES regions. Faces the shock sensor flags keep full dissipation.

With a DES-family hybrid. Without a RANS turbulence model every cell is treated as LES.

The blending function of Xiao et al. (2012), built from vorticity, strain rate, the turbulent length scale and the cell size, and combined with the shock sensor.

Scales the dissipation by the turbulence model's length-scale ratio, capped at 1, which is near 0 in LES regions and 1 in RANS regions. Faces the shock sensor flags keep full dissipation.

With a DES-family hybrid. Without a RANS turbulence model every cell is treated as LES.

No scaling: the low-dissipation fluxes apply full upwind dissipation.

The default.

The shock sensor is the normalised density jump between each cell and its face neighbours.

The default when unset.

The shock sensor is the Ducros ratio $|\nabla\cdot\mathbf{u}| / (|\nabla\cdot\mathbf{u}| + |\boldsymbol{\omega}|)$.

The same as velocity.

When an update leaves a cell's turbulence variables invalid, they are clipped to their lower limits.

When an update leaves a cell's turbulence variables invalid, the cell keeps its turbulence values from before the update.

The default when unset.