Central difference scheme

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Figure 8.2 Schematic of the backward, forward and central difference schemes.

The partial differential equations of the previous section are numerically solved. In order to minimize the number of spatial pivots required for a given accuracy, a central differences scheme was chosen. For certain flow problems the results may become unstable and the algorithm then allows for a smooth transition to one-sided differences with the penalty of an increased local error which has to be compensated by a larger number of pivots. 

In section 3.1.4, an analytical series solution using the matrizant was developed for the case where the coefficient matrix is a function of the independent variable. This methodology provides series solutions for Boundary value problems without resorting to any conventional series solution technique. In section 3.1.5, finite difference solutions were obtained for linear Boundary value problems as a function of parameters in the system. The solution obtained is equivalent to the analytical solution because the parameters are explicitly seen in the solution. One has to be careful when solving convective diffusion equations, since the central difference scheme for the first derivative produces numerical oscillations. 

Comparing (12.97) and (12.103), it is seen that the leading truncation error term in CDS is proportional to the square of the grid spacing, hence the central difference scheme is second order accurate both on uniform and non-uniform grids. 

The simplest TVD schemes are constructed combining the first-order (and diffusive) upwind scheme and the second order dispersive central difference scheme. These TVD schemes are globally second order accurate, but reduce to first order accuracy at local extrema of the solution. 

The convective terms were solved using a second order TVD scheme in space, and a first order explicit Euler scheme in time. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the smoothness monitor of van Leer [193] and the monotonic centered limiter [194]. The diffusive terms were discretized with a second order central difference scheme. The time-splitting scheme employed is of first order. 

The gradient terms were then approximated by the central difference scheme. For simplicity, the variables SrPN,SrSP, zPB, zwp, A, C, D and F are commonly introduced. The resulting equation can thus be written as ... 

The C-variables represent the convective fluxes through the grid cell surfaces and are normally approximated by the central difference scheme. 

The Tl-variables represent the generalized diffusion conductance and are related to the diffusive fluxes through the grid cell surfaces. In order to approximate these terms the gradients of the transported properties and the diffusion coefficients T are required. The property gradients are normally approximated by the central difference scheme. In a uniform grid the diffusion coefficients are obtained by linear interpolation from the node values (i.e., using arithmetic mean values) ... 

The the diffusion terms are discretized using the central-difference scheme. Radial direction ... 

To approximate scalar grid cell variables at the staggered w-velocity grid cell center node point, arithmetic interpolation is frequently used. The radial velocity component is discretized in the staggered t -grid cell volume and need to be interpolated to the w-grid cell center node point. The derivatives of the w-velocity component is approximated by a central difference scheme. When needed, arithmetic interpolation is used for the velocity components as well. 

To approximate the scalar variables at the staggered velocity grid cell surface points, arithmetic interpolation is frequently applied. The derivatives are approximated by a central difference scheme. 

The upwind scheme described here is first-order accurate in space while the central difference scheme is second-order accurate. Hence a central-difference scheme is preferred whenever possible. Since it is the grid Peclet number that decides the behavior of the numerical schemes, it is, in principle, possible to refine the grids until the grid Peclet is smaller than 2. This strategy, however, is often limited by the required computing time. With sufficiently fine meshes, the two schemes should give essen-... 

Both point-by-point and line-by-line overrelaxation methods were used to resolve the algebraic equations. ° An overrelaxation parameter of 1.5-1.8 was typically used. The two methods required similar computational times. An upwind scheme was used for all variables for high-Pe problems, while a central-difference scheme was used for low Pe. For some high-Pe cases, a central-difference scheme was used for the potential, but no appreciable differences in the results were observed. 

Coefficients a, a2 and b are obtained by the Fourier analysis and the relatively rapid solution of the resulting tridiagonal system of equations, due to the implicit nature of (2.31). A typical set is a = 22, a% = 1, and b = 24. To comprehend their function, let us observe Figure 2.2 that assumes the computation of dHy/dx and dEz/dx at i = 0. For the first case, constraint Ey = Ez = Hx = 0 at i = 0 indicates that dHy/dx (likewise for all H derivatives) must also be zero. In the second case, to calculate dEz/dx at i = one needs its values at i = —, . Nonetheless, point i = — is outside the domain and to find a reliable value for the tridiagonal matrix, the explicit, sixth-order central-difference scheme is selected... 

The second-order accurate central difference scheme is used in interior nodes, and the second-order accurate inward biased scheme is used on boundary nodes. Thus, the overall accuracy is kept second-order accurate in space. A staggered grid is used to compute the pressure at the cell center and velocities at surrounding grid lines. 

Again the leading term is the most significant, so for the central difference scheme. 

Following the procedure detailed in Chapter 3 for the case of unequal diffusion coefficients, with a non-uniform spatial grid, central difference scheme for the spatial derivative and the backward imphcit time integration, the coefficients for the Thomas algorithm for species y = A, B are given by... 

Equation 7 is a well-known expression for a second-order accurate estimation of the second-order derivatives using the central difference scheme. 

The central difference scheme can give rise to physically inconsistent solutions, in case IRI > 2. 

The CN method [20, 21] was used hy Randles [10] in his work carried out before the advent of modem computers. A central difference scheme is applied in which the diffusional term (10) is centered at the midpoint of the time-step... 

Solution of the Population Balance Equation for Liquid-Liquid Extraction Colunms using a Generalized Fixed-Pivot and Central Difference Schemes... 

Expressions (4a), (4b) and (4c) correspond to the so-called forward difference scheme, backward difference scheme and central difference scheme, respectively. The... 

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