Upwind differencing scheme

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

With standard DQMOM, when the explicit Euler scheme in time and the first-order upwind differencing scheme for space are employed, the volume-average weights in the cell centered at X at time (n + l)Af are... 

QUICK Differencing Scheme. The QUICK differencing scheme (Leonard and Mokhtari, 1990) is similar to the second-order upwind differencing scheme, with modifications that restrict its use to quadrilateral or hexahedral meshes. In addition to the value of the variable at the upwind cell center, the value from the next neighbor upwind is also used. Along with the value at the node P, a quadratic function is fitted to the variable at these three points and used to compute the face value. This scheme can offer improvements over the second-order upwind differencing scheme for some flows with high swirl. 

Integrating the generalized governing equation over this control volume and applying an upwind-differencing scheme, the discretized form of (10.1) becomes ... 

In addition to the central differencing and upwind differencing schemes, which are first-order schemes, another popular finite difference scheme is the QUICK scheme, a second-order upwind differencing scheme. Higher order means that more node points are involved when estimating the values of the dependent variables and their derivatives for formulating the finite difference equations. 

Shyy, W., Thakur, S. and Wright, J. (1992), Second-order upwind and central differencing schemes for recirculating flow computations, AIAA J., 30, 923-932. 

If a first-order upwind spatial-differencing scheme is used, the reconstructed moments tW/t,i i/2 at the cell interfaces are equal to the upwind values and, in the case of positive u, the following equation is obtained ... 

One of the most widely used difference-based VOF schemes is the high-resolution interface capturing (HRIC) scheme [21]. It is a normalized variable diagram (NVD) scheme based on a nonlinear blending of the bounded downwind (BD) and upwind differencing (UD) schemes, with the aim of combining the compressive property of the BD scheme with the stability of the UD scheme. 

Kietzmann et al. (1998) discussed discretization schemes for the solution of Eq. 8.25. The equation is a typical convection equation and an upwinding is required for numerical stability. However, the upwind discretization may result in numerical diffusion that blurs the flow front interface. A hybrid implicit scheme that combines upwind differencing and central differencing is therefore suggested. 

A Taylor series analysis on the ID transport equation shows that the transient artificial viscosity coefficients for explicit upwind differencing (i.e., a scheme that is 1. order in time and space) is given by [195, 196] ... 

Finite difference schemes based on the principles above are called central differencing . Central differencing schemes have stability problems in situations where convective transport is significant compared to the diffusive transport. Various methods have been devised to overcome this problem. The upwind scheme assigns the value of at a cell boundary to be the value at the node point from which the fluid is flowing, rather than the mean as in Eqs. (7.1.5) and (7.1.6). 

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