The gradient of a scalar property \(\phi\) is represented using the notation:

\[\grad \phi = \vec{e}_1 \frac{\partial}{\partial x_1} \phi + \vec{e}_2 \frac{\partial}{\partial x_2} \phi + \vec{e}_3 \frac{\partial}{\partial x_3} \phi\]

where the \(\vec{e}\) vectors represent the unit vectors of the 3-D space.


Gradient schemes are specified in the fvSchemes file under the gradSchemes sub-dictionary using the syntax:

    default         none;
    grad(p)         <optional limiter> <gradient scheme> <interpolation scheme>;


Gradient schemes

Interpolation schemes

  • linear: cell-based linear
  • pointLinear: point-based linear
  • leastSquares: Least squares

Gradient limiters

The limited gradient schemes attempt to preserve the monotonicity condition by limiting the gradient to ensure that the extrapolated face value is bounded by the neighbouring cell values.


See this example to see the relative performance of the schemes.

Further information

Source code: