Second-order accurate projective integrators for multiscale problems

We introduce new projective versions of second-order accurate Runge–Kutta and Adams–Bashforth methods, and demonstrate their use as outer integrators in solving stiff differential systems. An important outcome is that the new outer integrators, when combined with an inner telescopic projective integrator, can result in fully explicit methods with adaptive outer step size selection and solution accuracy comparable to those obtained by implicit integrators. If the stiff differential equations are not directly available, our formulations and stability analysis are general enough to allow the combined outer–inner projective integrators to be applied to legacy codes or perform a coarse-grained time integration of microscopic systems to evolve macroscopic behavior, for example.