Taking the Laplacian of a property \(\phi\) is represented using the notation:

\[\laplacian \phi = \frac{\partial^2}{\partial x_1^2} \phi + \frac{\partial^2}{\partial x_2^2} \phi + \frac{\partial^2}{\partial x_3^2} \phi\]

or as a combination of divergence and gradient operators

\[\div \left( \Gamma \grad \phi \right)\]

where \(\Gamma\) is a diffusion coefficient.

Usage🔗

Laplacian schemes are specified in the fvSchemes file under the laplacianSchemes sub-dictionary using the syntax:

laplacianSchemes
{
    default         none;
    laplacian(gamma,phi) Gauss <interpolation scheme> <snGrad scheme>
}

All options are based on the application of Gauss theorem, requiring an interpolation scheme to transform coefficients from cell values to the faces, and a surface-normal gradient scheme.

Example🔗

Further information🔗

• See the implementation details to see how the schemes are coded.