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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| Published in: | Computational mathematics and mathematical physics Vol. 60; no. 7; pp. 1134 - 1154 |
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| Format: | Journal Article |
| Language: | English |
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Moscow
Pleiades Publishing
01.07.2020
Springer Nature B.V |
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| ISSN: | 0965-5425, 1555-6662 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Kulikov, G. Yu |
| Author_xml | – sequence: 1 givenname: G. Yu surname: Kulikov fullname: Kulikov, G. Yu email: gkulikov@math.ist.utl.pt organization: CEMAT, Instituto Superior Técnico, Universidade de Lisboa |
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| CitedBy_id | crossref_primary_10_1016_j_dsp_2024_104941 crossref_primary_10_1016_j_jfranklin_2024_106768 crossref_primary_10_1134_S0965542523040115 crossref_primary_10_1016_j_apnum_2021_08_013 crossref_primary_10_1016_j_jfranklin_2022_11_021 crossref_primary_10_1016_j_ast_2024_109174 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Nested Implicit Runge–Kutta Pairs of Gauss and Lobatto Types with Local and Global Error Controls for Stiff Ordinary Differential Equations |
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