Upwind differencing

Boundary conditions may be written in terms of a known inlet pressure p 0) = po a vanishing velocity at the closed end U(L) = 0. Using these boundary conditions, formulate a discrete form of the governing equations. Be careful with the sense of the finite differences, considering the order of each derivative, upwind differencing, and boundary-condition information. 

A Taylor series analysis on the ID transport equation shows that the transient artificial viscosity coefficients for explicit upwind differencing is given by [157, 158] ... 

It can be noted that at least the explicit upwind method for the constant equation model gives the exact answer for CFL = 1, whereas the implicit upwind differencing method never does. The numerical viscosity of the implicit... 

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... 

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. 

We solve the partial differential equations using a finite difference method that employs the usual staggered grid arrangement. AU convective terms are modelled using upwind differencing for stability. As far as possible we use explicit methods, treating source terms implicitly where appropriate, to ensure that positive quantities remain positive. ... 

Equation (125) is different from the other equations in that it is one dimensional. It may be put into a form similar to Eq. (132) except that the terms involving derivatives with respect to are absent. It was discretized in tj using upwind differencing for the convective term and central differences for the other terms. The resulting tridiagonal matrix equation was directly solved. 

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 ... 

A uniform mesh is used with N elements where iV = 2 - -1. The level of refinement on a grid can then be described as grid level k refinement. The equations are discretised using first order upwind differencing, or second order central differencing where appropriate. This discretisation has been documented elsewhere [20] for the non thermal equations, so here we shall only consider the energy and two surface temperature equations. 

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. 

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