A Lyapunov exponents based stability theory for ODE initial value problem solvers

In this paper we consider the stability of variable step-size Runge-Kutta methods approximating bounded, stable, and time-dependent solutions of ordinary differential equation initial value problems. We use Lyapunov exponent theory to determine conditions on the maximum allowable step-size that guarantees the numerical solution of an asymptotically decaying time-dependent linear problem also decays. This result is used to justify using a one-dimensional asymptotically contracting real-valued nonautonomous linear test problem to characterize the stability of a Runge-Kutta method. The linear stability result is applied to show the stability of the numerical solution of stable nonlinear problems.