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

Abstract

This paper considers the mathematical framework of a sliced-time computation method for explosive solutions to systems of ordinary differential equations: Y(t)∈Rk:dYdt=F(Y), 0<t, Y(0)=Y0, that have finite or infinite explosion time. The method used generates automatically a sequence of non-uniform slices {[Tn−1,Tn]|n⩾1} determined by an end-of-slice condition that controls the growth of the solution within each slice. It also uses rescaling of the variables, whereas: t=Tn−1+βns and Y(t)=Y(Tn−1)+DnZn(s), Dn∈Rk×k, and βn being respectively an invertible diagonal matrix and a rescaling time factor. Thus, the original system is transformed into a sequence of slices-dependent initial-value shooting problems: dZnds=Gn(Zn), 0<s⩽sn, Zn(0)=0, ‖Zn(s)‖⩽S and ‖Zn(sn)‖=S, where S is a threshold value and ‖.‖ is the infinity norm on Rk. A suitable selection of βn and Dn leads the rescaled systems to satisfy a concept of uniform similarity, allowing to disable the extreme stiffness of the original ODE problem. Then, on each time slice, the uniformly rescaled systems are locally solved using a 4th order explicit Runge–Kutta scheme, within a computational tolerance of ϵloc. The sequential implementation of the local solver on a total of N slices leads to approximating the solution Y(t) of the original system within a global tolerance ϵglob.The proper definition of uniform similarity leads to deriving, under a stability assumption, a relationship between ϵloc, ϵglob and N. Such relation does not appear to be a sharp one particularly for the case when the existence time is infinite. In fact, numerical experiments conducted for infinite and finite times explosive discrete reaction diffusion problems attest for better estimates and for efficiency of the method in terms of stability and accuracy.