A globally adaptive explicit numerical method for exploding systems of ordinary differential equations (Q1946149)

This paper considers the mathematical framework of a sliced-time computation method for non-oscillatory blow-up problems, specifically systems of ordinary differential equations whose solution has strictly positive components, a monotonously increasing L-infinity norm and an explosive behavior. The method generates automatically a sequence of non-uniform time slices resulting in a sequence of initial value shooting problems that are rescaled on the basis of a change of variables. The uniform similarity of these problems allows for similar numerical simulations on all time slices avoiding some issues resulting from solving extremely stiff and exploding systems directly. The authors prove, under a stability assumption, a bound for the global growth in terms of the local growth and the number of slices. Such a relation is shown to be suboptimal, particularly, when the existence time is infinite which is the case for the first numerical test. Another test with finite blow-up time is also shown to demonstrate the extreme stability of the method.