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Open Access 2024 | Open Access | Buch

Solving Ordinary Differential Equations in Python

verfasst von: Joakim Sundnes

Verlag: Springer Nature Switzerland

Buchreihe : Simula SpringerBriefs on Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Inhaltsverzeichnis
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SUCHEN

Über dieses Buch

Dieses Open-Access-Buch erklärt die Grundlagen moderner Löser für gewöhnliche Differentialgleichungen (ODEs). Die Formulierung und Lösung von ODEs ist ein wesentlicher Bestandteil der mathematischen Modellierung und Computerwissenschaft, und zahlreiche Lösungsansätze sind in kommerzieller und Open-Source-Software verfügbar. Allerdings ist kein einzelner ODE-Löser für jedes einzelne Problem die beste Wahl, und die Auswahl des richtigen Lösers erfordert grundlegende Einsichten in die Funktionsweise der Löser. Dieses Buch wird genau diese Einsicht bieten, um Studenten und Forschern zu ermöglichen, den richtigen Löser für jedes ODE-Problem auszuwählen, das von Interesse ist, oder, falls nötig, ihre eigenen Löser zu implementieren. Die Präsentation ist kompakt und zugänglich und konzentriert sich auf die große und weit verbreitete Klasse von Lösern, die als Runge-Kutta-Methoden bekannt sind. Explizite und implizite Methoden werden motiviert und erklärt, ebenso wie Methoden zur Fehlerkontrolle und automatischen Zeitschrittauswahl, und alle Löser werden in Python als Klassenhierarchie implementiert.
Mit KI übersetzt

Inhaltsverzeichnis

Frontmatter
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Open Access

Chapter 1. Programming a Simple ODE Solver
Abstract
Ordinary differential equations (ODEs) are widely used in science and engineering, particularly when it comes to modeling dynamic processes. Although analytical methods can be employed to solve simple ODEs, nonlinear ODEs typically require numerical methods for solutions. In this chapter we demonstrate how to program general numerical solvers capable of handling any ODE. Initially we will focus on scalar ODEs, which consist of a single equation and a single unknown. Subsequently, in Section 1.3, we will extend these concepts to systems of coupled ODEs. Acquiring a solid grasp of the concepts presented in this chapter will not only help you with programming your own ODE solvers but also in using a diverse range of readily available, general-purpose ODE solvers in Python or other programming languages.
Joakim Sundnes
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Open Access

Chapter 2. Improving the Accuracy
Abstract
As mentioned previously, the FE method derived in Chapter 1 is not the most sophisticated ODE solver. Although it provides sufficient accuracy for most of the applications covered in this book, there are alternative methods available that offer improved accuracy and stability, making them better suited for solving challenging ODE systems. In this chapter, we will focus on enhancing accuracy, and thus we will primarily explore explicit methods. Implicit methods, which exhibit superior stability properties and are better suited for solving stiff ODEs, will be discussed in Chapter 3.
Joakim Sundnes
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Open Access

Chapter 3. Stable Solvers for Stiff ODE Systems
Abstract
In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.
Joakim Sundnes
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Open Access

Chapter 4. Adaptive Time Step Methods
Abstract
In practical computations, one seeks to achieve a desired accuracy with the minimum computational effort. For a given method, this requires finding the largest possible value of the time step Δt. In the previous chapters we kept the step size constant through the solution interval, but this is rarely the most efficient approach, since the error depends on the characteristics of the solution in addition to the step size. In smooth regions, larger time steps can be used without introducing significant error, while smaller time steps are needed in regions where the solution has rapid variations. This chapter extends the Runge-Kutta methods from the previous chapters to methods that select the time step automatically to control the error in the solution.
Joakim Sundnes
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Open Access

Chapter 5. Modeling Infectious Diseases
Abstract
Throughout this book we have focused entirely on solving ODEs, without delving deeply into their origins or applications. In the present chapter we shift our attention to modeling with ODEs, by exploring a widely studied class of ODE models that describe the spread of infectious diseases. This group of models serves as a good example of how a complex phenomenon can be modeled using relatively simple systems of ODEs. We will derive these models based on a set of fundamental assumptions and discuss the limitations that arise from these assumptions. Although we consider a single application and a single class of models, the fundamental steps of the modeling process are applicable to a wide range of real-world phenomena.
Joakim Sundnes
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Backmatter
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Metadaten
Titel
Solving Ordinary Differential Equations in Python
verfasst von
Joakim Sundnes
Copyright-Jahr
2024
Electronic ISBN
978-3-031-46768-4
Print ISBN
978-3-031-46767-7
DOI
https://doi.org/10.1007/978-3-031-46768-4

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