Scaling of Differential Equations

Book Description

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.

This open book is licensed under a Creative Commons License (CC BY-NC). You can download Scaling of Differential Equations ebook for free in PDF format (6.4 MB).