Matrix-Free Iterative Processes for Implementation of Implicit Runge–Kutta Methods
In this work we present so-called generalized Picard iterations (GPI) – a family of iterative processes which allows to solve mildly stiff ODE systems using implicit Runge–Kutta (IRK) methods without storing and inverting Jacobi matrices. The key idea is to solve nonlinear equations arising from the base IRK method by special iterative process based on the idea of artificial time integration. By construction these processes converge for all asymptotically stable linear ODE systems and all A-stable base IRK methods at arbitrary large time steps. The convergence rate is limited by the value of “stiffness ratio”, but not by the value of Lipschitz constant of Jacobian. The computational scheme is well suited for parallelization on systems with shared memory. The presented numerical results exhibit that the proposed GPI methods in case of mildly stiff problems can be more advantageous than traditional explicit RK methods.
KeywordsRunge–Kutta methods Stiff problems Parallel methods
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