✍ Scribed by Laurent O. Jay; Thierry Braconnier
No coin nor oath required. For personal study only.
The main di culty in the implementation of most standard implicit Runge-Kutta (IRK) methods applied to (sti ) ordinary di erential equations (ODEs) is to e ciently solve the nonlinear system of equations. In this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit Euler method. The preconditioner is based on the W-transformation of the RK coe cient matrices discovered by Hairer and Wanner. For sti ODEs the preconditioner is by construction asymptotically exact for methods with an invertible RK coe cient matrix. The methodology is particularly useful when applied to super partitioned additive Runge-Kutta (SPARK) methods. The nonlinear system can be solved by inexact simpliÿed Newton iterations: at each simpliÿed Newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system.
📜 SIMILAR VOLUMES
For implicit Runge-Kutta methods intended for stiff ODEs or DAEs, it is often difficult to embed a local error estimating method which gives realistic error estimates for stiff/algebraic components. If the embedded method's stability function is unbounded at z = o0, stiff error components are grossl