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  • Book
  • Open Access
  • © 2016

Scaling of Differential Equations

  • No comparable book exists

  • Scaling is a classical topic in applied mathematics, but here strongly connected to numerical simulations

  • The book contains a wide range of examples, of differing complexity, from many different scientific fields

  • Includes supplementary material: sn.pub/extras

Part of the book series: Simula SpringerBriefs on Computing (SBRIEFSC, volume 2)

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Table of contents (4 chapters)

  1. Front Matter

    Pages i-xiii
  2. Dimensions and units

    • Hans Petter Langtangen, Geir K. Pedersen
    Pages 1-15Open Access
  3. Ordinary differential equation models

    • Hans Petter Langtangen, Geir K. Pedersen
    Pages 17-68Open Access
  4. Basic partial differential equation models

    • Hans Petter Langtangen, Geir K. Pedersen
    Pages 69-98Open Access
  5. Advanced partial differential equation models

    • Hans Petter Langtangen, Geir K. Pedersen
    Pages 99-134Open Access
  6. Back Matter

    Pages 135-138

About this book

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.

Authors and Affiliations

  • Center for Biomedical Computing, Simula Research Laboratory, Fornebu, Norway

    Hans Petter Langtangen

  • Department of Mathematics, University of Oslo, Oslo, Norway

    Geir K. Pedersen

About the authors

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 published over 100 scientific publications and written several books, including papers and the bestseller TCSE 6 "A Primer on Scientific Programming with Python", now in its 5th edition. He has also developed open source and commercial software systems for computational sciences.

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.     

Bibliographic Information

Buy it now

Buying options

Softcover Book USD 24.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access