Efficient implementation of Radau collocation methods
L. Brugnano, F. Iavernaro, C. Magherini
TL;DR
The paper addresses the computational bottleneck of solving implicit Runge-Kutta systems for stiff IVPs by introducing an augmented low-rank Radau IIA formulation with auxiliary abscissae. This yields a 2s-stage representation where a Crout-based splitting can be tuned so that a single $LU$ factorization of an $m\times m$ system suffices per inner iteration, achieving $O(m^3)$-level cost per step in favorable cases. Linear convergence analysis using the test equation shows the proposed splitting attains $L$-convergence and favorable amplification factors compared to prior schemes, enabling efficient outer iterations with few inner steps. Numerical tests on several stiff problems demonstrate improved efficiency for larger systems (with 2–3 inner iterations), while small problems may require different inner-iteration choices; overall, the method offers a practical, extendable path to faster Radau IIA implementations and to other implicit RK/collocation methods. The work has potential impact on stiff ODE solvers by reducing inner solve costs and enabling scalable, sequential-line implementations.
Abstract
In this paper we define an efficient implementation of Runge-Kutta methods of Radau IIA type, which are commonly used when solving stiff ODE-IVPs problems. The proposed implementation relies on an alternative low-rank formulation of the methods, for which a splitting procedure is easily defined. The linear convergence analysis of this splitting procedure exhibits excellent properties, which are confirmed by its performance on a few numerical tests.