Hans Petter Langtangen & Svein Linge 
Finite Difference Computing with PDEs [EPUB ebook] 
A Modern Software Approach

This book is open access under a CC BY 4.0 license.This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.