Switching algorithm

We are now ready to discuss the treatment of differential equations with switching conditions. The main task is the localization of the switching points as roots of the switching functions. Note, the complexity of this task depends on the form of the switching function. Switching functions may in general have the same complexity as the right hand side function. For example, an impact requires the computation of relative distances for both functions. 

After every integration step check if a discontinuity has occurred by testing the sign of the switching function. 

If no sign changed, continue the integration otherwise localize the discontinuity as a root of the switching function. 

Change to the new right hand side, modify variables according to Eq. (6.2.1). 

These steps are described in detail in the following sections. 

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Goal 4. Provide chronotropic support and mode-switching algorithms as required. 

There are a variety of ways in which the hop can be calculated. The most commonly used variant is Tully s least switches algorithm [73]. This is designed to reproduce the correct electronic state populations with the least number of hops. The probability of changing from electronic state 2 to 1 is here given by... 

However, for the use in switching algorithms, see Sec. 6, the interpolation error is propagated as it influences the determination of the switching point and the point of restart. Thus in this case, the interpolation error must be controlled. In [SG84] it is shown for Adams methods that the continuous representation is error controlled, for BDF methods the corresponding result can be found in [EichQl]. 

It should be noted that the way an Adams method is implemented influences the continuous representation of the solution. Special care has to be taken in P EC) implementations, see Sec. 4.1.1. In that case X (l) a n+i- For this type of implementation (4.5.1) deflnes no continuous representation of the solution. Using (4.5.1) in that case may lead to problems when passing from one interval to the next in connection with switching algorithms, which will be described in Sec. 6. 

Sometimes discontinuities occur at previously known time points and the integration can be stopped and restarted at these time events after a re-initialization. In other cases, the occurrence of a discontinuity is state dependent and one attempts to describe it implicitly in terms of roots of so-called switching functions. Integration methods must be extended to localize these roots in order to permit a re-initialization also at these points. For the numerical solution of discontinuous systems we present a switching algorithm, the basic principles of which are given in Sec. 6.3. 

Note that the figure implies that the solution for the old right hand side can be continued, at least for a small interval. This assumption is necessary for switching algorithms. 

Now we have discussed all steps of the switching algorithm. The overall algorithm is summarized in Fig. 6.3. 

The realization in a program can be done with the help of an additional memo-vector, which corresponds to a. In the right hand side it must be read-only . It is allowed to be changed only by the switching algorithm after a root of of change has been localized. 

If i thresh crosses One of the threshold values Si this leads to a discontinuity. The treatment with standard switching algorithms involves the computation of all n/b + l switching functions... 

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