Laplacian schemes

Table of Contents

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.


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

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.


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