ODEs with Discontinuities

The equations of motion of multibody systems often contain discontinuities in the right hand side, in the solution or in its higher derivatives. In the sequel we summarize discontinuities in the function itself as well as in its higher derivatives under the expression discontinuity . 

In multibody systems typical elements introducing discontinuities are  

We saw in Ch. 4 that numerical integration methods require continuity also of higher derivatives of the solution depending on the order of the method and the error control mechanism. Ignoring this fact causes wrong error estimates and a failure of step size and error control. Thus, standard numerical integration software shows unreliable and inefficient behavior when applied to non-smooth systems. We 

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. 

This method is of a particular importance in the context of optimization methods, where the computation of sensitivity matrices with prescribed accuracy is required, see Sec. 7. 

---

The difficulties arising in the integration of ODEs with discontinuities are often circumvented by using integration methods with fixed step size or low order. The first approach has the drawback that the error is not controlled. Both approaches have the drawback that they may be very inefficient. 

For the numerical treatment of ODEs with discontinuities it is necessary to rewrite the differential equation as... 

E1181] Ellison D. (1981) Efficient automatic integration of ODEs with discontinuities. Math. Comput. Simul. 23(1) 12-20. 

---