Half-Explicit Methods

All methods for DAEs we discussed so far were implicit. A nonlinear constraint fc defines the algebraic variables in an implicit way. Therefore, iteration methods, mainly of Newton type are inevitable when treating DAEs. On the other hand, the differential part of the problem may be discretized by an explicit scheme. Such a method is called a half-explicit method. 

We introduce half-explicit methods by considering the explicit Euler method for index-1 and index-2 problems. 

Unfortunately, most explicit Runge-Kutta formulas have very poor convergence properties when used for half-explicit methods in the above way. In general they do not exceed order two. Exceptions are those sets of coefficients developed in [BH93]. We cite the coefficients of their third order method... 

Example 5.4.3 The 5 order Dormand and Prince method can be extended to an half-explicit method. For this method s = 6 and the coefficients aij,bi, i,j — 1,... 6 are given in (4-2.11). The additional coefficients can be selected in various ways. Arnold ([Am98]) sets them 50, that the residual of the hidden constraint ( -constraint) is damped in an optimal way. This leads to the coefficients of the code... 

Bra92] Brasey V. (1992) A half-explicit method of order 5 for solving constrained mechanical systems. Computing 48 191-201. 

Fig. 9 Different descriptions of the metal-metal bond orders in some dinuclear complexes with bridging hydride ligands according to the electron-counting method. The half-electron method does not explicitly take into account the 3-center-2-electron nature of the M-H-M interaction and thus results in a greater M-M bond order than would be predicted theoretictilly. In contrast, the half-arrow method treats the 3-center-2-electron nature of the M-H-M interactions explicitly and thereby predicts an M-M bond order which is in accord with theory...

For an index-1 problem the half-explicit Euler method reads... 

Multistep generalizations of this approach have first been introduced and discussed by Arevalo [Are93], extrapolation methods based on this idea were introduced by Lubich [Lub91] and resulted in the code MEXX [LENP95]. Ostermann investigated half-explicit extrapolation methods for index-3 problems [Ost90]. 

We will restrict us here to the presentation of half-explicit one-step methods. They have been first introduced in [HLR89]. 

Half-Explicit Runge-Kutta Methods for Index-1 Problems... 

The half-explicit Euler method can be extended in various ways to get higher order methods. In the index-1 case such an extension is... 

For the index-2 a half-explicit Runge-Kutta method can be defined by... 

Arn98] Arnold M. (1998) Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT, Numerical Analysis, to appear. 

Ost90] Ostermann A. (1990) A half-explicit extrapolation method for differential-algebraic systems of index 3. IMA J, Numer, Anal, 10 171-180. 

In the ion-electron method of balancing redox equations, an equation for the oxidation half-reaction and one for the reduction half-reaction are written and balanced separately. Only when each of these is complete and balanced are the two combined into one complete equation for the reaction as a whole. It is worthwhile to balance the half-reactions separately since the two half-reactions can be carried out in separate vessels if they are suitably connected electrically. (See Chap. 14.) In general, net ionic equations are used in this process certainly some ions are required in each half-reaction. In the equations for the two half-reactions, electrons appear explicitly in the equation for the complete reaction—the combination of the two half-reactions—no electrons are included. 

From this starting point, the authors develop equations leading to the evaluation of the real symmetric K matrix to specify the scattering process on the repulsive surface and propose its determination by a variational method. Furthermore, they show explicitly the conditions under which their rigorous equations reduce to the half-collision approximation. A noteworthy result of their approach which results because of the exact treatment of interchannel coupling is that only on-the-energy-shell contributions appear in the partial linewidth. Half-collision partial linewidths are found not to be exact unless off-the-shell contributions are accidentally zero or (equivalently) unless the interchannel coupling is zero. The extension of the approach to indirect photodissociation has also been presented. The method has been applied to direct dissociation of HCN, DCN, and TCN and to predissociation of HCN and DCN (21b). 

In the ADI method [4-6], the explicit and implicit methods are combined. The timesteps, AT, are divided into two half-timesteps, ATf2. For the first of these half-timesteps, the derivatives along the. -coordinate... 

There is a surprise here, however. The Feldberg school uses a trick that renders the simple start highly accurate. At each step n, the time corresponding to that step, which ought perhaps to be nSt, is instead corrected by half a time interval to (n — l/2)St. This is the same device previously used for the explicit box method, and described as a fudge in Sect. 1.3.7. In this case, however, there is rational basis for doing this, as was shown recently [9]. Briefly, it turns out... 

In many simple chemical equations, such as Equation 20.2, balancing the electrons is handled automatically —that is, we balance the equation without explicitly accounting for the transfer of electrons. Many redox equations are more complex than Equation 20.2, however, and cannot be balanced easily without taking into account the number of electrons lost and gained. In this section, we examine the method of half-reactions, a systematic procedure for balancing redox equations. 

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