Implicit methods improvements

The truncation errors in (5.9) and (5.12) are of the same magnitude, but the implicit Euler method (5.11) is stable at any positive step size h. This conclusion is rather general, and the implicit methods have improved stability properties for a large class of differential equations. The price we have to pay for stability is the need for solving a set of generally nonlinear algebraic equations in each step. 

Because iteration is required for the implicit method, it takes more computational resources to solve the same problem using the same value of the time step. However, because of its significantly improved stability properties, the implicit method can take a much larger time step than the explicit method. Therefore there is the potential for the implicit method to be more efficient, as long as it can use larger time steps and still maintain sufficient accuracy. 

An improvement over the explicit method is done by the fully implicit method in which the concentration corresponding to time (j + 1) is calculated as... 

Unlike explicit methods, the performance of implicit methods cannot be simply judged by conventional statistical measures such as goodness of fit. As pointed out in the literature,18 spurious effects such as system drift and covariations among constituents can be incorrectly interpreted as arising from the analyte of interest. This scenario has led to the development of hybrid methods in which elements of explicit and implicit techniques are combined to improve performance. 

Both the explicit and implicit methods already described have been improved to greater accuracy and, hopefully, greater efficiency. 

A perhaps more interesting method is that of Saul yev [496] (and apparently independently, the same idea, of Barakat [70] a short time later). The method is explicit, which makes programming easier than implicit methods, and is capable of improvements over the original idea. There are two basic variants that make up the building blocks for improvements. The LR variant, as the name implies moves from left (that is, from X = 0) to right (higher X), generating new values at the next time level. The computational molecule for this is... 

The alternating direction implicit (ADI) method (Peaceman and Rachford, 1955) is a partially implicit method. The equation is rearranged so that one coordinate may be solved implicitly using the Thomas algorithm whilst the others are treated explicitly. If this is done alternately, each coordinate has a share of the implicit iterations and the efficiency (Gavaghan and Rollett, 1990) as well as the stability is improved. The method was used by Heinze for microdisc simulations (Heinze, 1981 Heinze and Storzbach, 1986) and has subsequently been adopted by others (Taylor et al, 1990 Fisher et al., 1997). 

For the free energy of solvation calculation, however, it is difficult to discern the most accurate method. Recently, there have been numerous publications exploring the use of the cluster continuum method with anions. With regard to implicit solvation, there are no definite conclusions to the most accurate method, yet for the PB models the conductor-like models (COSMO CPCM) appear to be the most robust over the widest range of circumstances [23]. At this writing, the SMVLE method seems to be the most versatile, as it can be used by itself, or with the implicit-explicit model, and the error bars for bare and clustered ions are the smallest of any continuum solvation method. The ability to add explicit water molecules to anions and then use the implicit method (making it an implicit-explicit model) improves the results more often than the other implicit methods that have been used in the literature to date. 

An improved simulation might therefore be obtained by using an estimate of the average concentration value during the period Af. This is done in the implicit method, which considers the previous data point as well as the next, yet to be determined point in the computation. In fact, there are many different implicit methods. Here we only illustrate the simplest of them, which assumes that all variables change linearly over a sufficiently small interval A f. 

Comparison of the results for asimu aexatl obtained here with those of the explicit method of section 9.2 indicates that the implicit method significantly improves the accuracy of the computation. Moreover, by using (9.3-5) to squeeze a number of small steps in one row, we can further reduce the error, which now goes as (At)2. This is illustrated in Fig. 9.3-1, which compares the results obtained for the first- order reaction A —> with = 1, k = 1,... 

Michelsen s third order semi-implicit Runge-Kutta method is a modified version of the method originally proposed by Caillaud and Padmanabhan (1971). This third-order semi-implicit method is an improvement over the original version of semi-implicit methods proposed in 1963 by Rosenbrock. 

In practice, implicit multistep methods are used to improve upon approximations obtained by explicit methods. This combination is the so-called predictor-corrector method. Predictor-corrector methods employ a single-step method, such as the Runge-Kutta of order 4, to generate the starting values to an explicit method, such as an Adams-Bashforth. Then the approximation from the explicit method is improved upon by use of an implicit method, such as an Adams-Moulton method. Also, there are variable step size algorithms associated with the predictor-corrector strategy in the literature [5,25]. 

The implicit multistep methods add stability but require more computation to evaluate the implicit part. In addition, the error coefficient of the Adams-Moulton method of order k is smaller than that of the Adams Bashforth method of the same order. As a consequence, the implicit methods should give improved accuracy. In fact, the error coefficient for the imphcit fourth-order Adams Moulton method is 19/720, and for the explicit fourth-order Adams Bashforth method it is 251/720. The difference is thus about an order of magnitude. Pairs of exphcit and implicit multistep methods of the same order are therefore often used as predictor-corrector pairs. In this case, the explicit method is used to calculate the solution,, at v +i. Furthermore, the imphcit method (corrector) uses y + to calculate /(x +i,y +i), which replaces /(x +i,y +i). This allows the solution, y +i, to be improved using the implicit method. The combination of the Adams Bashforth and the Adams Moulton methods as predictor orrector pairs is implemented in some ODE solvers. The Matlab odel 13 solver is an example of a variable-order Adams Bashforth Moulton multistep solver. 

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. 

Janezic, D., Orel, B. Implicit Runge-Kutta Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 33 (1993) 252-257 Janezic, D., Orel, B. Improvement of Methods for Molecular Dynamics Integration. Int. J. Quant. Chem. 51 (1994) 407-415... 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

Since Sg satisfies a system of equations identical with that satisfied by X, except that Rg replaces H, a standard method for improving the result of a direct method is to do the replacement and solve as before for the correction. This obtained correction will not, of course, be the true Sg) but only an approximation Sg, and it will have been obtained as a result of a set of operations that is equivalent to the formation of CaRg, in accordance with Eq. (2-3). The Ca = C is not known explicitly, but is defined implicitly (see the methods of factorization below). 

---