Scaling of Differential Equations

This volume in the Simula SpringerBriefs series gives thorough explanations, and several examples, of how to reason about finding the right scales in numerical simulations.


The Simula SpringerBriefs on Computing

 aims to provide introductions to select areas of research in computing. Each volume aims to provide an introduction to, and overview over, research fields that can otherwise be inaccessable. Scaling of Differential Equations by prof. Hans Petter Langtangen and prof. Geir K. Pedersen is the second volume in this series.

The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling.  A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models.

Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics.

The text is easily accessible and example-driven. The first part on ODEs fits even a lower undergraduate level, while the most advanced multiphysics fluid mechanics examples target the graduate level. The scientific literature is full of scaled models, but in most of the cases, the scales are just stated without thorough mathematical reasoning. This book explains how the scales are found mathematically.

This book will be a valuable read for anyone doing numerical simulations based on ordinary or partial differential equations.

The book is available from Springer Publishing. The official electronic version is open access, and also available from the Authors web site. 

About the Authors
Hans Petter Langtangen

Hans Petter Langtangen is a professor of computer science at the University of Oslo. He has formerly been a professor of mechanics and is now the director of a Norwegian Center of Excellence: "Center for Biomedical Computing", at Simula Research Laboratory. Langtangen has written several books, and published over 100 scientific publications on topics ranging from physics and medicine to design of software systems. He has also developed open source and commercial software systems for computational sciences.

Geir K. Pedersen

Geir K. Pedersen is a professor of mechanics at the Department of Mathematics, University of Oslo. He has a life-long experience in fluid dynamics and mathematical modeling. Pedersen has published articles on wave theory, numerical modeling, perturbation techniques, tsunamis, hydrodynamic stability and experimental fluid dynamics.