NVD and TVD schemes

Many of the convection schemes in OpenFOAM are based on the Normalised Variable (NV) [44] and Total Variation Diminishing (TVD) [25] methods. These provide a set tools to characterise scheme properties such as boundedness (1-D).

- Offer a blend between a low order scheme and a higher order scheme based on the calculation of a limiter
- Boundedness of NVD/TVD schemes is only guaranteed for 1-D cases
- Boundedness can be improved in 2-D and 3-D cases by limiting the gradient

For arbitrary unstructured meshes it is not straightforward to identify the upstream and downstream locations reliably. The limiters are evaluated using a compact stencil as described in Jasak et al. [28], where:

TVD: Limiter \(\Psi(r)\) defined as a function of \(r\)

\[ r = 2 \frac{\vec{d} \dprod \left( \grad{\phi} \right)_P}{\vec{d} \dprod \left(\grad{\phi} \right)_f} - 1 = 2 \frac{\vec{d} \dprod \left( \grad{\phi} \right)_P}{\phi_N - \phi_P} - 1 \]

NVD: Limiter \(\widetilde{\phi_f}\) defined as a function of \(\widetilde{\phi_c}\)

\[ \widetilde{\phi_c} = 1 - 0.5 \frac{\vec{d} \dprod \left( \grad{\phi} \right)_f}{\vec{d} \dprod \left( \grad{\phi}\right)_P } = 1 - 0.5 \frac{\phi_N - \phi_P}{\vec{d} \dprod \left(\grad{\phi}\right)_P} \]

Where the gradient at cell P \( \left( \grad{\phi} \right)_P \) is calculated using the user-selected gradient scheme, and the vector \( \vec{d} \)

\[ \vec{d} = \vec{C}_{N} - \vec{C}_{P} \]

Source code:

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