Hans Petter Langtangen & Geir K. Pedersen 
Scaling of Differential Equations [PDF ebook] 

The book serves both as a reference forvarious scaled models with corresponding dimensionless numbers, and as aresource for learning the art of scaling. A special feature of the book is the emphasis on how to create softwarefor scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is amathematical technique that greatly simplifies the setting of input parameters innumerical simulations. Moreover, scaling enhances the understanding of howdifferent physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequentlyencountered, but very often in only a brief and shallow setting, the presentbook gives much more thorough explanations of how to reason about finding theright scales. This process is highly problem dependent, and therefore the bookfeatures a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics.The text is easily accessible andexample-driven. The first part on ODEs fits even a lower undergraduate level, while the most advanced multiphysics fluid mechanics examples target thegraduate level. The scientific literature is full of scaled models, but in mostof the cases, the scales are just stated without thorough mathematicalreasoning. This book explains how the scales are found mathematically.This book will be a valuable read for anyonedoing numerical simulations based on ordinary or partial differential equations.