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__One-sided numerical methods to locate event points in discontinuous ODEs__

L. Lopez and L. Dieci

**Abstract**

We present a numerical approach to treat discontinuous differential systems
of ODEs of the type: $x'=f_1(x)$ when $h(x)<0$ and $x'=f_2(x)$ when $h(x)>0$,
and with $f_1\neq f_2$ for $x\in \Sigma$,
where $\Sigma:=\{x: h(x)=0\}$ is a smooth co-dimension one
discontinuity surface.
Often, $f_1$ ($f_2$) cannot be evaluated when $h(x)>0$ ($h(x)<0$) and
for this reason, we consider numerical schemes
which do not require $f_1$ above $\Sigma$ (respectively, $f_2$ below $\Sigma$).
The use of explicit schemes allows us to avoid the evaluation of the numerical solution
in the forbidden region (see [1, 2]) but can produce oscillations of the numerical solution around the event point on the discontinuity surface. A penalty technique, for the stepsize, can produce a sequence of numerical solutions which approach $\Sigma$ from one-side only.
We remark that in this talk we restrict attention only to accurately locate the event point where a trajectory reaches $\Sigma$.

**Bibliography**

[1]
L. Dieci, L. Lopez,
Numerical Solution of Discontinuous Differential
Systems: Approaching the Discontinuity Surface from
One Side, App. Num. Math, (2011), In Press.

[2]
L. Dieci, L. Lopez,
A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side, J. Comput. Appl. Math, vol. 236 (16), (2012), pp 3967-3991.