Implicit Schemes

So far, we have considered the explicit scheme of finite-difference formulations and its stability criterion for an illustrative example. The use of the explicit scheme becomes somewhat cumbersome when a rather small Ax is selected to eliminate the truncation error for accuracy. The Ai allowed then by the stability criterion may be so small that an enormous amount of calculations may be required. We now intend to eliminate this difficulty by giving different forms to the equations resulting from the finite-difference formulation. Let us take the case of one-dimensional conduction in unsteady problems, for which we obtained the difference equation given by Eq. (4.50). Consider a formulation of the problem in terms of backward rather than forward differences in time. That is, decrease the time from f +i = (n + l)Ai to tn = nAt. Thus we obtain 

Subtracting from this relation its initial steady value given by Eq. (4.53) yields 

Since all differences in Eq. (4.73) are positive according to Fig. 4.20,3 this equation is satisfied under all circumstances. Therefore, it is unconditionally stable. We obtain this stability, however, at the cost of a new algebraic complexity. Recall Eq. (4.50), in which all temperatures except T[ +l are known, and the latter is obtained by solution of the equation. By contrast, in Eq. (4.72) only T is known, and the application of this equation to the nodes yields a tridiagonal matrix which then can be solved by the method discussed in Ex. 4.1. 

Reconsider Ex. 4.7. We wish to determine the transient temperature distribution of the flat plate by using an implicit numerical scheme. 

In view of Eq. (4.72), we can write a matrix representation for each inner node as 

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A reasonable approach for achieving long timesteps is to use implicit schemes [38]. These methods are designed specifically for problems with disparate timescales where explicit methods do not usually perform well, such as chemical reactions [39]. The integration formulas of implicit methods are designed to increase the range of stability for the difference equation. The experience with implicit methods in the context of biomolecular dynamics has not been extensive and rather disappointing (e.g., [40, 41]), for reasons discussed below. 

Fig. 1. A semi-implicit integrator the implicit scheme is applied only to the fast system.

Equation (2.106) gives rise to an implicit scheme except for 0 = 0. The application of implicit schemes for transient problems yields a set of simultaneous equations for the field unknown at the new time level n + 1. As can be seen from Equation (2.111) some of the terms in the coefficient matrix should also be evaluated at the new time level. Therefore application of the described scheme requires the use of iterative algorithms. Various techniques for enhancing the speed of convergence in these algorithms can be found in the literature (Pittman, 1989). 

Common practice involves also the implicit scheme... 

In dealing with the pure implicit scheme we might have... 

In order to prove this theorem, we beforehand reduce the implicit scheme (la) to the explicit scheme (32) with the operator C —... 

The implicit scheme Byo Ay = Q with any operator B = B > 0 is also absolutely unstable. 

Afterwards when the sweep formulae became customary, one began to analyze in full details two-layer implicit schemes (weighted schemes) for which R = crA. These schemes obviously represent a particular case of the scheme with R = (tAq. 

The case cr yt 0, relating to the implicit scheme, is connected with the equation related to the unknown y =... 

The following implicit schemes are frequently encountered in the theory and practice ... 

Summarizing, the weighted scheme (9) is stable in the space C, provided condition (17) holds. For the purely implicit scheme with [Pg.467]

In this chapter the new difference schemes are constructed for the quasilin-ear heat conduction equation and equations of gas dynamics with placing a special emphasis on iterative methods available for solving nonlinear difference equations. Among other things, the convergence of Newton s method is established for implicit schemes of gas dynamics. 

For this, it seems unreasonable to employ explicit schemes with fastly varying ingredients k(u), c(u) and f(u). The power functions of temperature reflect in full measure the diiflculties involved in such a case. For any implicit scheme one possible stability condition... 

Various implicit schemes for the quasUinear heat conduction equation. 

Some consensus of opinion is desirable in this matter, since a smaller number of operations is performed in the explicit scheme, but it is stable only for sufficiently small values of r. In turn, the implicit scheme being absolutely stable requires much more arithmetic operations. 

What schemes are preferable for later use Is it possible to bring together the best qualities of both schemes in line with established priorities In other words, the best scheme would be absolutely stable as the implicit schemes and schould require in passing from one layer to another exactly Q arithmetic operations. As in the case of the explicit schemes, Q would be proportional to the total number of the grid nodes so that Q = 0 l/hf). 

In what follows one possible example demonstrates for a system of ordinary differential equations that there is an implicit scheme which is rather economical than the explicit ones requiring the additional operations. 

Recall that in the case of the one-dimensional heat conduction equation a similar implicit scheme is associated on every layer with the difference... 

Here the passage from the jth layer to the j + l)th layer is carried out in the following two steps the first one involves the explicit scheme and the second one - the implicit scheme as suggested above. 

Seidel method. As we have mentioned above, implicit schemes are rather stable in comparison with explicit ones. Seidel method, being the simplest implicit iterative one, is considered first. The object of investigation here is the system of linear algebraic equations... 

As a matter of fact, the upper relaxation method and Seidel method are nothing more than the implicit scheme (6) with B E incorporated. Still using the usual framework of implicit iterative methods, the value yk+i is determined from the equation... 

The main result in Section 2 regarding the optimal set of parameters tj, can be generalized directly for implicit schemes with B E as follows ... 

From what has been said above another conclnsion can be drawn in this direction the po.ssible applications of the implicit scheme (6) in solving the original equation Au = / are equivalent to the numerical. solution of the auxiliary equation Cv = through the use of the explicit scheme... 

Recall that Theorem 2 has been proved in Section 2 placing a special emphasis on one particular case of explicit schemes with the identity operator B = E involved. To make our exposition more transparent, the implicit scheme transforms into the explicit scheme (17). Having stipulated the conditions y E < C < estimate... 

What can happen upon replacing the describing explicit scheme by an implicit one Being concerned with one of the implicit schemes... 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

For practical purposes, implicit schemes are the methods of choice when the solution is smooth and well behaved as a function of time. In that case much larger time steps can be taken than with explicit schemes, thus allowing a reduction in the computational effort. When large temporal gradients and rapid variations are expected, accuracy constraints set severe limits to the time-step size. In that case explicit schemes might be favorable, as they come with a reduced numerical effort per time step. 

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