Optimally zero stable explicit peer methods with variable nodes
type: Item
instance of: scholarly article
title: Optimally zero stable explicit peer methods with variable nodes
zbMATH Open document ID: 1432.65089
author: Marcel Klinge
author: R. Weiner
author: H. Podhaisky
published in: BIT
publication date: 2018-06-26T00:00:00Z
review text: The authors develop explicit peer-to-peer methods for solving unstable initial problems, in which some of the values of the stage are copies of the values of the stage from the previous steps. They argue that this reduces the number of function calls per step and can be interpreted as a generalization of the ``first as last'' principle, known for Runge-Kutta methods, which states that the last stage is evaluated at the same point as the first stage of the next stage. \par The article presents a special class of explicit peer-to-peer methods with shifted stages. This allows one to save the estimates of the functions. In addition, the constructed methods are superconvergent and converge faster than expected. \par The authors focus on optimally zero stable methods, which give excellent results. The order conditions are also derived and it is proved that under some natural conditions numerical approximations can be calculated for all step-size sequences. Concrete optimally zero stable methods are constructed up to order 8. \par At the end of the work, some numerical experiments are given. They confirm that the new methods are more effective than the built-in functions of Matlab, such as ode23, ode45 and ode113.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 6894727
zbMATH Keywords: explicit peer methods
zbMATH Keywords: variable nodes
zbMATH Keywords: step size control
zbMATH Keywords: Runge-Kutta methods
reviewed by: Georgiĭ Anatol'evich Sviridyuk
describes a project that uses: Matlab
describes a project that uses: ode23s
describes a project that uses: MATLAB ODE suite
describes a project that uses: ode23
describes a project that uses: ode45
describes a project that uses: Ode15s
describes a project that uses: ode113
MaRDI profile type: Publication
full work available at URL: s10543-017-0691-8
OpenAlex ID: W2769254558
cites work: A 3(2) pair of Runge-Kutta formulas
cites work: Numerical Methods for Ordinary Differential Equations
cites work: On the derivation of explicit two-step peer methods
cites work: Q4875914
cites work: A family of embedded Runge-Kutta formulae
cites work: Solving Ordinary Differential Equations I
cites work: Strong stability preserving explicit peer methods
cites work: Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control
cites work: The MATLAB ODE Suite
cites work: Two-step peer methods with continuous output
cites work: Explicit two-step peer methods
cites work: Superconvergent explicit two-step peer methods
DOI: 10.1007/S10543-017-0691-8
description: scientific article; zbMATH DE number 6894727