Differencing schemes

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain 

Example 8.12 Use the backward differencing method to solve the heat transfer problem of Example 8.3. Select A-fc = 0.25 and Aj = 0.0625. 

This system of equations is solved for each beginning with the inlet boundary  

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S.A. Silling, Stability and Accuracy of Differencing Schemes for Viscoplastic Models in Wavecodes, SAND91-0141, Sandia National Laboratories, Albuquerque, NM, 1991. 

In order to increase the accuracy of the approximation to the convective term, not only the nearest-neighbor nodes, but also more distant nodes can be included in the sum appearing in Eq. (37). An example of such a higher order differencing scheme is the QUICK scheme, which was introduced by Leonard [82]. Within the QUICK scheme, an interpolation parabola is fitted through two downstream and one upstream nodes in order to determine O on the control volume face. The un-... 

The QUICK scheme has a truncation error of order h. However, similarly as in the case of the central differencing scheme, at high flow velocities some of the coupling coefficients of Eq. (37) become negative. 

In order to minimize numerical diffusion, Boris and Book [131] formulated the idea of blending a low-order stable differencing scheme with a higher order, potentially unstable, scheme in such a way that steep concentration gradients are maintained as well as possible. The algorithm they proposed consists of the following steps ... 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

Chow, L.C. and Tien, C.L. An Examination of Four Differencing Schemes for Some Elliptic-Type Convection Equations , Numerical Heat Transfer, Vol. 1, 1978, pp. 87-100. 

Interpolation with Eq. (6.25) is second-order accurate. This approximation is called the central differencing scheme (CDS). 

It can be seen that the central differencing scheme discussed above is conservative. The coefficients of CDS satisfy the Scarborough criterion. However, for uniform grid (Ae = 0.5), when the Peclet number is higher than 2, the coefficients oe will become negative [Fg > This violates the boundedness requirements and may... 

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

FIGURE 7.12 Results of VOF simulations at 0.12 s (influence of differencing scheme), (a) Power law. (b) QUICK (SUPERBEE). 

On a Cartesian grid, n = x at the e face and the usual schemes can be used to estimate gradient at e. For example, a central differencing scheme will give ... 

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. 

SOM Second Order Moments SOR Successive Over-Relaxation SUPERBEE SUPERBEE function TDM A Tri-Diagonal Matrix Algorithm TVD Total Variation Diminishing UDS Upstream Differencing Scheme... 

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 ... 

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... 

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