Gradient schemes

Table of Contents

• Usage
• Options
  • Gradient schemes
  • Interpolation schemes
  • Gradient limiters
• Example
• Further information

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.

Usage

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

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

Options

Gradient schemes

• Gauss gradient scheme
• Least-squares gradient scheme

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.

• Cell-limited gradient scheme
• Multi-directional cell-limited gradient scheme
• Face-limited gradient scheme
• Multi-directional face-limited gradient scheme
• clippedLinear: limits linear scheme according to a hypothetical cell size ratio

Example

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

Further information

Source code:

• Foam::fv::gradScheme

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