The Fully Implicit Scheme

The fully implicit scheme is based on the second-order accurate Adams-Moulton formulae, that is, at each time step, n + 1, all terms (both linear and nonlinear) in Eqs. (1.11)-(1.13) are integrated implicitly as follows  

Since the governing equations are nonlinear, the implementation of the implicit algorithm requires an iterative method. A direct iterative scheme based on Newton s method is computationally too demanding. Instead, as an alternative, a predictor-corrector scheme can be used with the corrector to be iteratively applied until a convergence criterion is met. To this end, in Eqs. (1.19) and (1.20) + Peff, 

The advantage of the full implicit scheme and the direct enforcement of the continuity equation can be more clearly seen when we formulate a suitable Poisson equation for the pressure by taking the divergence of the momentum equation, Eq. (1.2)  

With the semi-implidt/explicit scheme, the terms in the first parentheses of the right-hand side of Eq. (1.18) are not identically zero. However, with the full implicit scheme in which the continuity equation is satisfied with machine accuracy, these terms are zero and therefore the periodic part of the pressure is more accurately evaluated. 

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The truncation error for Eq. (10.40) is the same as those stated for the momentum equation for p = 0, A, 1. The fully implicit scheme can be increased to a formal second-order accuracy by representing the streamwise derivatives with three-level (i-l,i,i+l) second-order differences. For any implicit method, the finite difference momentum and energy equations are algebraically nonlinear in the unknowns because of the quantities unknown at the i+1 level in the coefficients. Linearizing procedures can and have been used, but are beyond the scope of this book. 

The Crank-Nicolson method is a mixture of the implicit and explicit schemes which is less stable but more accurate than the fully implicit scheme (see Britz, 1988). A set of simultaneous equations analogous to those necessary for the fully implicit method must be solved the matrix [Af] has exactly the same structure in either case. 

Eirst, we should comment on the semi-implidt/explicit and fully implicit schemes. The fully implicit scheme gives more physical results and is more accurate and more... 

Figure 1.9 Comparison between the semi-implicit/explicit and the fully implicit scheme for the rms vorticity fluctuations. (Adapted from Ref [55].)...

Stable allowing considerably larger time steps of integration to be used. On the other hand, it requires both more computational memory and more arithmetic operations per time step. A comparison between these two schemes is offered in Figure 1.9, in which the average vorticity components are given as a function of the distance from the wall. It is clearly seen there that the fully implicit scheme with At = 10 and mesh size 96 x 97 x 96 yields the same statistics with the semi-implidt/explidt scheme with At = 2 x 10 and mesh size 144 x 145 x 144. 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Implicit schemes are unconditionally stable, this is shown in Fig. 8.27 where the evolution of the temperature, in a/ag steps, for values of a/ag higher than 0.5 is shown. Higher values of a/ag mean that we can use higher At, which at the end implies lower computational cost and faster solutions. The results in Fig 8.27 were obtained with the fully implicit Euler scheme, i.e., to = 1. The comparison between the implicit Euler and the Crank-Nicholson, to = 0.5 is illustrated in Fig. 8.28 for the center line temperature evolution. Although there is no apparent significance difference, we expect that the CN scheme is more accurate due to its second order nature. 

The backward differencing method requires the solution of 7+1 simultaneous equations to find the radial temperature profile. It is semi-implicit since the solution is still marched-ahead in the axial direction. Fully implicit schemes exist where (7-I- l)(7-l-1) equations are solved simultaneously, one for each grid point in the total system. Fully implicit schemes may be used for problems where axial diffusion or conduction is important so that second derivatives in the axial direction, or 9 r/9z, must be retained in the partial differential equa-... 

Beckermann and coworkers [42,43] discretized (10.29) using a control volume-based FDM, in which the transient term is treated by a fully implicit scheme. The resulting algebraic equation in cartesian coordinates can be expressed as ... 

Classification of Simulation Methods by Time Stepping Scheme Commercial flow simulators generally discretise time derivatives using a first order finite difference formula (Euler s method). The time derivative thus involves the difference of functions at the end and at the start of each time step. All other terms in the equations are discretised to involve functions evaluated at the start and the end of each time step. The pressure always appears at the end of the time step and one says that the pressure is implicit. Saturations appear at the end of the time step in the fully implicit approach. The saturation... 

There are many variants on these ideas. For example, one can form a scheme in the pressure-saturation formulation where the pressure is imphcit, as usual, but in the pressure equation the saturation is evaluated at the start of the time step, and in the saturation equation the scheme is fiiUy implicit in all variables. This gives improved stability compared to the IMPES scheme, but it is not as stable as the fully implicit method. 

The coupled set of flow and solid equations (Eqs. (3.1)-(3.8), (3.11)) were solved simultaneously. A finite volume scheme was adopted for the spatial discretization of the flow equations and solution was obtained with a SIMPLER method for the pressure-velocity field [8]. In the case of transient simulation, the 1-D transient solid energy equation was solved with a second order accurate, fully implicit scheme by using a quadratic backward time discretization [9]. The coupled flow and solid phases were solved iteratively and convergence was achieved at each time step when the solid temperamre did not vary at any position along the wall by more than 10 K. 

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

The governing equations of the model are discretized in space by means of the finite element method [3, 18], and in time through a fully implicit finite difference scheme (backward difference) [18], resulting in the nonlinear equation set of the following form, [4, 7],... 

The equation for the central point (i = 1) actually plays the role of inner boundary condition. The above system should be completed with one more boundary condition for the outer point tm = R. Irrespective of the type of the used time difference scheme (explicit, fully implicit or Crank-Nicholson), the further treatment of the resulting system of difference equations is absolutely analogous to the one developed for Cartesian coordinates. 

The equation is solved by means of iteration and fully-implicit finite difference in the following scheme (Wen, 1997b),... 

The temperature equation is solved separately, whereas the water mass conservation and momentum conservation equations are solved together. Because of the strong non-linearity that are present in these equations, a fully implicit time integration scheme is used. 

The partial differential equations are solved numerically with a fully implicit numerical scheme embodied in PHOENICS code [9]. This code solves following general differential equations,... 

Looking at Figure 3, considering time on the abscissa (lower scale), f(xQ - /i)= a,- (old value) and f(xQ + /i) = a (new value), the chord AB approximates the slope at A, Le. at a time r, according to a forward difference scheme (classic explicit method) on the other hand, it constitutes a central difference approximation at a time t + 0.5Ar (Crank-Nicolson). We can also use it as a backward difference approximation for the slope at B, Le, at a time t + At (Laasonen, fully implicit method) [4,6] ... 

Finite Volume Methods The finite volume method, when the permeabihty tensor is diagonal in the selected coordinate system, approximates the pressure and saturation functions as piecewise constant in each grid block. The flux components are assumed constant in their related half-cells. Thus when two cells are joined by a face, the related component of flux is assumed to be the same each side of the face. The balance laws are invoked separately on each grid block, and are discretised in time either by an explicit or fully implicit first order Euler scheme or other variant as discussed in the previous subsection. 

Another way to overcome the step-size restriction fc < is to use multiple-time-stepping methods [4] or implicit methods [17, 18, 12, 3). In this paper, we examine the latter possibility. But for large molecular systems, fully implieit methods are very expensive. For that reason, we foeus on the general class of scmi-implicit methods depicted in Fig. 1 [12]. In this scheme. Step 3 of the nth time step ean be combined with Step 1 of the (n - - l)st time step. This then is a staggered two-step splitting method. We refer to [12] for further justification. 

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