Nested Implicit Runge–Kutta Pairs of Gauss and Lobatto Types with Local and Global Error Controls for Stiff Ordinary Differential Equations

The problem of efficient global error estimation and control is studied in embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types as applied to stiff ordinary differential equations (ODEs). Stiff problems may arise in many areas of engineering, and their accurate numerical solution is...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics Vol. 60; no. 7; pp. 1134 - 1154
Main Author: Kulikov, G. Yu
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.07.2020
Springer Nature B.V
Subjects:
Applications of mathematics
Computational Mathematics and Numerical Analysis
Differential equations
Error analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Numerical integration
Ordinary differential equations
Runge-Kutta method
Solvers
automatic local and global error controls
embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types
absolute and scaled local and global error estimates
ordinary differential equation
stiff problem
ISSN:0965-5425, 1555-6662
Online Access:Get full text
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Summary:The problem of efficient global error estimation and control is studied in embedded nested implicit Runge–Kutta pairs of Gauss and Lobatto types as applied to stiff ordinary differential equations (ODEs). Stiff problems may arise in many areas of engineering, and their accurate numerical solution is an important issue of computational and applied mathematics. A cheap global error estimation technique designed recently for the mentioned Runge–Kutta pairs can severely overestimate the global error when applied to stiff ODEs and, hence, this reduces the efficiency of those solvers. In the present paper, we explain the cause of that error overestimation and show how to improve the mentioned computation techniques for stiff problems. Such modifications not only boost the efficiency of numerical integration of stiff ODEs, but also make the embedded nested implicit Runge–Kutta pairs with scaled modified local and global error controls superior to stiff built-in MATLAB ODE solvers with only local error control when applied to important test examples.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542520070076