A Matlab package for automatically generating Runge-Kutta trees, order conditions, and truncation error coefficients

Authors:
Frank Cameron
Affiliations:
Tampere University of Technology, Finland, Pori, Finland
Venue:
ACM Transactions on Mathematical Software (TOMS)
Year:
2006

Citing 13
Cited 3

Implicit Runge-Kutta methods for differential algebraic equations

SIAM Journal on Numerical Analysis
Introduction to algorithms

Introduction to algorithms
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems

Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Coefficients of the Taylor expansion for the solution of differential-algebraic systems

Applied Numerical Mathematics
Symbolic derivation of Runge-Kutta methods

Journal of Symbolic Computation
A new recurrence for computing Runge-Kutta truncation error coefficients

SIAM Journal on Numerical Analysis
High-order explicit Runge-Kutta pairs with low stage order

Applied Numerical Mathematics - Special issue celebrating the centenary of Runge-Kutta methods
On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs

SIAM Journal on Numerical Analysis
Stage Value Predictors and Efficient Newton Iterations in Implicit Runge--Kutta Methods

SIAM Journal on Scientific Computing
Formal series and numerical integrators, Part II: application to index-2 differential-algebraic systems

proceedings of the on Numerical analysis of hamiltonian differential equations
IRK methods for DAE: starting algorithms

Proceedings of the on Numerical methods for differential equations
The Design and Analysis of Computer Algorithms

The Design and Analysis of Computer Algorithms
Symbolic derivation of Runge-Kutta order conditions

Journal of Symbolic Computation

Letter to the Editor: Enumeration results for Runge-Kutta methods for index 2 DAEs

Journal of Computational and Applied Mathematics
Symbolic derivation of order conditions for hybrid Numerov-type methods solving y˝=f(x,y)

Journal of Computational and Applied Mathematics
Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations

Journal of Computational Physics

Quantified Score

Hi-index	0.01

Visualization

Abstract

In designing parts of Runge-Kutta methods, order conditions and truncation error coefficients (TECs) are needed. Order conditions and TECs are typically presented as a set of trees combined with rules for producing algebraic expressions from the trees. The tree sets are defined recursively and can be generated by hand only for low orders. This article describes a package of Matlab routines for automatically generating Runge-Kutta trees, order conditions, and TECs. The routines are capable of generating Maple code, Matlab code, or LaTeX expressions for ODEs or DAEs of index 1 and 2. In producing the package, two theoretical problems are tackled: (a) avoiding the repeated generation of the same tree and (b) the efficient storage of TECs.