document: The extrapolation of the upwind value required for TVD differencing is a particular hurdle for the application on unstructured meshes. As discussed in Section 3.2, two methods to extrapolate the value at the virtual upwind node, using data readily available on unstructured meshes, are considered. Given how the virtual upwind node is incorporated in the gradient ratio rf, the extrapolation method of Darwish and Moukalled [13] is referred to as implicit extrapolation and the method introduced by Ubbink and Issa [12] as explicit extrapolation. Both methods precisely reconstruct the upwind value for equidistant, rectilinear meshes but fail to do so on non-equidistant or non-rectilinear meshes, as discussed in Section 3.2. Using the explicit extrapolation method this issue can be rectified by imposing appropriate limits on the extrapolated upwind value.
doc_bio_tags: O B I I I I O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O B I I I I I O O O O O B I I I I I I I I I O O O O B I I O O O O O B I O O B I I I I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O
extractive_keyphrases: darwish and moukalled data readily available on unstructured meshes explicit extrapolation extrapolation of the upwind value implicit extrapolation method introduced by ubbink and issa tvd differencing unstructured meshes virtual upwind node is incorporated in the gradient ratio rf
abstractive_keyphrases: equidistant, rectilinear meshes non-equidistant non-rectilinear meshes