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NVD and TVD schemes

Many of the convection schemes in OpenFOAM are based on the Normalised Variable (NV) 40 and Total Variation Diminishing (TVD) 23 methods. These provide a set tools to characterise scheme properties such as boundedness (1-D).

NVD
TVD

Properties​

  • 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

Limiter calculation​

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. 26, where:

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

r=2d ∙ (∇ϕ)Pd ∙ (∇ϕ)f−1=2d ∙ (∇ϕ)PϕN−ϕP−1 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 ϕf~\widetilde{\phi_f} defined as a function of ϕc~\widetilde{\phi_c}

ϕc~=1−0.5d ∙ (∇ϕ)fd ∙ (∇ϕ)P=1−0.5ϕN−ϕPd ∙ (∇ϕ)P \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 (∇ϕ)P\left( \grad{\phi} \right)_P is calculated using the user-selected gradient scheme, and the vector d\vec{d}

d=CN−CP \vec{d} = \vec{C}_{N} - \vec{C}_{P}

Further information​

Source code:

API: