An Exit Hole method for Verified Solution of IVPs for ODEs using Linear Programming for the Search of Tight Bounds

In his survey [5], Nedialkov stated that ?Although high-order Taylor series may be reasonably efficient for mildly stiff ODEs, we do not have an interval method suitable for stiff ODEs.? This paper is an attempt to find such a method, based on building a positively invariant set in extended state space. A positively invariant set is treated as geometric generalization of differential inequalities. We construct a positively invariant set from simpler sets which are not positively invariant, but have exit hole instead. The exit holes of simpler sets are suppressed during the construction. This paper considers only sets which are polytopes. Linear interval forms are used to evaluate a projection of ODE velocity vector to the normals of the polytope facets. This permits the use of Linear Programming for the search of tighter positively invariant set. The Exit Hole method is illustrated by stiff Van der Pol ODE.