Preservation of the growth rates of delay differential equations by Euler schemes with non-uniform step sizes

In this paper, we consider explicit Euler methods which recover the rate of growth to infinity of a highly nonlinear autonomous delay differential equation. The success of the methods rely on the step size changing in response to the state, as it has been shown that Euler methods with constant step size will systematically underestimate the growth rate. It is also shown that the computed solution converges to the true solution on any compact time interval, when a parameter which controls the step size is sent to zero. A second method, which applies to a related ordinary differential equation, and which requires less computational effort, is also presented.