Numerical Methods for Ordinary Differential Equations

Kees Vuik; Fred Vermolen; Martin van Gijzen; Thea Vuik

doi:10.5074/t.2023.001

Numerical Methods for Ordinary Differential Equations

Authors

Kees Vuik, Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands; Fred Vermolen, Department of Computational Mathematics, Faculty of Sciences, University of Hasselt, Belgium; Martin van Gijzen, Delft Institute of Applied Mathematics (DIAM), Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands; Thea Vuik, VORtech, The Netherlands

DOI: https://doi.org/10.5074/t.2023.001

Keywords: numerical methods, differential equations, numerical integration, interpolation, nonlinear equations

Synopsis

In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize the aspects that play an important role in practical problems. We conﬁne ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation. The techniques discussed in the intro-ductory chapters, for instance interpolation, numerical quadrature and the solution to nonlinear equations, may also be used outside the context of differential equations. They have been in-cluded to make the book self-contained as far as the numerical aspects are concerned. Chapters, sections and exercises marked with a * are not part of the Delft Institutional Package.
The numerical examples in this book were implemented in Matlab, but also Python or any other programming language could be used. A list of references to background knowledge and related literature can be found at the end of this book. Extra information about this course can be found at http://NMODE.ewi.tudelft.nl, among which old exams, answers to the exercises, and a link to an online education platform. We thank Matthias Moller for his thorough reading of the draft of this book and his helpful suggestions.

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Author Biographies

Kees Vuik, Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands

Prof.dr.ir. C. (Kees) Vuik is a Full Professor in Numerical Analysis at the Delft Institute of Applied Mathematics of the TU Delft in The Netherlands. He obtained his PhD degree from Utrecht University in 1988. Thereafter he joined the TU Delft as an assistant professor in the section Numerical Analysis. His research is related to the discretization of partial differential equations, moving boundary problems, High-Performance Computing, and iterative solvers for large linear systems originating from incompressible fluid flow; wave equations; and energy networks. He has teached the numerical analysis course for more than 30 years.

Fred Vermolen, Department of Computational Mathematics, Faculty of Sciences, University of Hasselt, Belgium

Prof.dr.ir. F.J. (Fred) Vermolen is a Full Professor in Computational Mathematics at the University of Hasselt in Belgium. He obtained his PhD degree from the TU Delft in 1998. Thereafter he worked at CWI and from 2000 he joined the TU Delft as an assistant professor in the section Numerical Analysis. His research is related to analysis, numerical methods and uncertainty quantification for partial differential equations. He has given courses in numerical analysis for more than 10 years.

Martin van Gijzen, Delft Institute of Applied Mathematics (DIAM), Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands

Prof. dr. M.B. (Martin) van Gijzen is Full Professor in High-Performance Computing at the Delft Institute of Applied Mathematics of the TU Delft in The Netherlands. He obtained his PhD degree from the same university in 1994. Before returning to the TU Delft in 2004, he hold positions at the Utrecht University and at TNO, both in The Netherlands, and at CERFACS in France. His research topics include High Performance Computing, iterative solvers for large scale linear systems and eigenvalue problems, model order reduction, and inverse problems. He has worked on many applications, including topology optimisation, MR imaging, numerical modelling of ocean circulation, and acoustic and elastic wave propagation.

Thea Vuik, VORtech, The Netherlands

Thea Vuik got her PhD from the Numerical Analysis group at Delft University of Technology. In addition to her research on troubled-cell indication in DG methods, she assisted in the Numerical Analysis course for several years. Currently, she is working at VORtech as a scientific software engineer.