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We investigate a high-order, fully explicit, asymptotic-preserving scheme fora kinetic equation with linear relaxation, both in the hydrodynamic anddiffusive scalings in which a hyperbolic, resp. parabolic, limiting equationexists. The scheme first takes a few small (inner) steps with a simple,explicit method (such as direct forward Euler) to damp out the stiff componentsof the solution and estimate the time derivative of the slow components. Theseestimated time derivatives are then used in an (outer) Runge-Kutta method ofarbitrary order. We show that, with an appropriate choice of inner step size,the time-step restriction on the outer time step is similar to the stabilitycondition for the limiting macroscopic equation. Moreover, the number of innertime steps is also independent of the scaling parameter. We analyse stabilityand consistency, and illustrate with numerical results.

A high-order asymptotic-preserving scheme for kinetic equations using projective integration