Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations
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Abstract
A new class of stable methods for solving ordinary differential equations (ODEs) is introduced. This is based on combining the Local Linearization (LL) integrator with other extant discretization methods. For this, an auxiliary ODE is solved to determine a correction term that is added to the LL approximation. In particular, combining the LL method with (explicit) Runge Kutta integrators yields what we call LLRK methods. This permits to improve the order of convergence of the LL method without loss of its stability properties. The performance of the proposed integrators is illustrated through computer simulations.
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Authors and Affiliations
Universidad de Granma, Bayamo, MN, Cuba
H. De la Cruz
Universidad de las Ciencias Informáticas, La Habana, Cuba
H. De la Cruz
Instituto de Cibernética, Matemática y FÃsica, La Habana, Cuba
R. J. Biscay, F. Carbonell & J. C. Jimenez
Institute of Statistical Mathematics, Tokyo, Japan
T. Ozaki
- H. De la Cruz
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- R. J. Biscay
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- F. Carbonell
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- J. C. Jimenez
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- T. Ozaki
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Editors and Affiliations
Advanced Computing and Emerging Technologies Centre, The School of Systems Engineering, University of Reading, RG6 6AY, Reading, United Kingdom
Vassil N. Alexandrov
Department of Mathematics and Computer Science, University of Amsterdam, Kruislaan 403, 1098, Amsterdam, SJ, The Netherlands
Geert Dick van Albada
Faculty of Sciences, Section of Computational Science, University of Amsterdam, Kruislaan 403, 1098, Amsterdam, SJ, The Netherlands
Peter M. A. Sloot
Computer Science Department, University of Tennessee, 37996-3450, Knoxville, TN, USA
Jack Dongarra
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De la Cruz, H., Biscay, R.J., Carbonell, F., Jimenez, J.C., Ozaki, T. (2006). Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations. In: Alexandrov, V.N., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science – ICCS 2006. ICCS 2006. Lecture Notes in Computer Science, vol 3991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758501_22
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DOI: https://doi.org/10.1007/11758501_22
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Keywords
- Phase Portrait
- Local Linearization
- Matrix Exponential
- General Linear Method
- Local Linearization Approximation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
