On the AN-Stability of Runge–Kutta Methods Combined with Richardson Extrapolation

Richardson extrapolation (RE) is an adaptive numerical technique that computes a linear combination of numerical solutions produced by using the same base method with different time-step sizes in order to improve the accuracy. A qualitative property that we expect from the resulting combined method when stiff problems are to be solved is A-stability, which provides a bounded numerical solution on Dahlquist’s autonomous test problem by any choice of the time-step size. Conditions of A-stability have been extensively studied for certain combinations of Runge–Kutta (RK) methods with several versions of RE. However, the A-stability of a base method is not automatically inherited by the combined method. Moreover, A-stability has some limitations when a non-autonomous initial-value problem is to be solved, e.g., a problem obtained by semi-discretization of a time-dependent PDE. Therefore, a possible generalization of A-stability called AN-stability was advocated. Our aim is to study the conditions under which an RK method combined with RE results in an AN-stable method. Simultaneously, an easily verifiable algebraic condition is gained for A-stability.