Multistep Algorithms

In many batch applications, scheduling and coordination of the beginning and end periods of steps within the production process are required. With modern control algorithms, multistep processes can be designed where production rates can... 

Remapping methods must also be able to move material into neighbor eells that share only a node. This is ealled eorner eoupling and is required to aeeurately move material diagonally aeross the mesh. For example, the flow shown in Fig. 9.15 is at a 30° angle and material from element zero will flow into elements one, two, and three. Elements one and two share a faee with element zero. But element three shares only a node with element zero. The multistep algorithm moves material from element zero to element one, then from element one to element three when the direetion is switehed. 

Term (2) calculated by the high-accuracy multistep CBS-APNO method (Section 5.5.2.2b) was 341.2 kcal mol-1 or 1426 kJ mol-1. The Sackur-Tetrode equation for the gas-phase entropy of the proton was mentioned in this regard, but in fact the algorithm automatically handles this. 

In 50 the author presents a further investigation of the frequency evaluation techniques which are recently proposed by Ixaru et al. for exponentially fitted multistep algorithms for the solution of first-order ordinary differential equations (ODEs). These studies have a scope which is to maximize the benefits of the exponentially-fitted methods via the evaluation of the frequency of the problem. The proposed by Ixaru and co-workers method for frequency... 

Once a model is constructed, it can be used for the optimization of all time and length scales, using a similarly constructed multistep algorithm (see Figure 4.11). The multistep algorithms in Figures 4.11 and 4.12 are the same numerically, but with... 

Single step methods, such as the Eulerian forward and backward algorithms, can be regarded as particular cases of multistep methods (Byrne and Hindmarsh, 1975). One of them, developed by Gear (1971) is particularly well adapted to stiff systems. In this algorithm, the solution to (5.2) is approximated by... 

For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980 s by Flindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG. 

The two equations, [53] and [55], form a system of coupled ODEs with the variable z playing the role of the independent variable. Given initial conditions at a point Zq these equations can be solved by standard numerical routines such as those discussed in the previous section. Because much computational effort is required to evaluate each p, at each increment of the independent variable z, a method that does not require too many evaluations of the right hand side of the iterative equation is desirable. Usually, a simple forward Euler routine is quite adequate for these purposes. If a multistep algorithm is used, the Adams-Bashforth method has been recommended by Kubicek and Marek the first-order Adams-Bashforth algorithm is, in fact, equivalent to the simple forward Euler algorithm. 

A multistep algorithm uses y or/and y obtained in the previous points, tn-i to calculate y +p... 

The third-order exphcit multistep algorithm by Adams-Bashforth ... 

One-step algorithms are always stable for h 0. In the case of multistep algorithms, the stability problem is further complicated by the presence of parasite solutions, which can prevail on the real solution and make the algorithm unstable even for h 0. 

---