The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas

DOI10.1016/0168-9274(94)00031-XzbMath0817.65060OpenAlexW2040434486MaRDI QIDQ1339338

Syvert P. Nørsett, Anne Kværnø, Kenneth R. Jackson

Publication date: 1 December 1994

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0168-9274(94)00031-x

zbMATH Keywords

numerical examplesButcher seriessimple iterationB-seriesorder of accuracymodified Newton iterationorder of errordifferential-algebraic systems of index 1implicit Runge-Kutta algorithm

Mathematics Subject Classification ID

Nonlinear ordinary differential equations and systems (34A34) Numerical methods for initial value problems involving ordinary differential equations (65L05) Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations (65L06) Error bounds for numerical methods for ordinary differential equations (65L70)

Related Items

Are high order variable step equistage initializers better than standard starting algorithms?, Parallelizing spectral deferred corrections across the method, PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method, B-series analysis of iterated Taylor methods, IRK methods for DAE: Starting algorithms, Runge-Kutta research at Toronto, CWI contributions to the development of parallel Runge-Kutta methods, An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values, Construction of starting algorithms for the RK-Gauss methods

Cites Work

• Unnamed Item
• Unnamed Item
• Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations
• A method for constructing generalized Runge-Kutta methods
• The numerical solution of differential-algebraic systems by Runge-Kutta methods
• An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values
• The Modified Newton Method in the Solution of Stiff Ordinary Differential Equations
• Implicit Runge–Kutta Methods for Differential Algebraic Equations
• Embedded Diagonally Implicit Runge-Kutta Algorithms on Parallel Computers
• The error committed by stopping the Newton iteration in the numerical solution of stiff initial value problems
• A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear systems of ordinary differential equations