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This text presents the theory of finite difference schemes applied to nonlinear partial differential difference equations with small dissipation. It is designed to be used as an introductory text for students in applied mathematics, engineering, and the sciences, and with that in mind, presents the theory of finite difference schemes in a way that is both rigorous and accessible to the typical graduate student in the course. The two aims of the text are to present the basic material necessary to do scientific computation with finite difference schemes and to present the basic theory for understanding these methods.