An analysis of the order of Runge-Kutta methods that use an iterative scheme to compute their internal stage values

DOI10.1007/BF01733789zbMath0864.65051OpenAlexW2070495144MaRDI QIDQ2565272

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

Publication date: 11 March 1997

Published in: BIT (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf01733789

zbMATH Keywords

numerical examplesNewton iterationorderimplicit Runge-Kutta methods\(B\)-seriesinternal stage values

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)

Related Items

The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas, Third- and fourth-order ESDIRK methods for stiff and differential-algebraic problems, Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems, B-series analysis of iterated Taylor methods, Performance of Gauss implicit Runge-Kutta methods on separable Hamiltonian systems., Runge-Kutta research at Toronto, Runge-Kutta research in Trondheim, Formal series and numerical integrators. I: Systems of ODEs and symplectic integrators, Construction of starting algorithms for the RK-Gauss methods

Uses Software

• SIMPLE

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Parallel methods for initial value problems
• Parallel step-by-step methods
• The search for the Holy Grail, or: Predictor-corrector methods for solving ODEIVPs
• Efficient block predictor-corrector methods with a small number of corrections
• Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations
• Order conditions for Rosenbrock type methods
• The error behaviour of a general class of predictor-corrector methods
• On the Butcher group and general multi-value methods
• The use of Butcher series in the analysis of Newton-like iterations in Runge-Kutta formulas
• Parallel iteration of high-order Runge-Kutta methods with stepsize control
• An implementation of singly-implicit Runge-Kutta methods
• 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
• The Potential for Parallelism in Runge–Kutta Methods. Part 1: RK Formulas in Standard Form
• Numerical Solution of Ordinary Differential Equations