Generalized Picard iterations: A class of iterated Runge-Kutta methods for stiff problems

A family of iterated Runge-Kutta (RK) methods capable of handling both non-stiff and stiff problems with complex spectrum is developed. The methods are based on classical implicit RK methods, are matrix-free and have natural parallelization capability. They are equivalent to first-order explicit RK schemes with unlimited number of stages. It is shown that they converge to the solution which corresponds to the underlying implicit RK method for all asymptotically stable linear problems y′=Jy with arbitrary big time steps τ. The convergence rate is limited by the 'stiffness ratio' ρ(J)ρ(J−1), rather than τρ(J). The experiments show that the proposed methods outperform explicit Dormand-Prince DOP853 code in stiff case and can compete with it in non-stiff case.