Chapter 2: The Finite Difference Method
The chapter presents the Finite Difference Method (FDM). This method dates back to Euler1 who introduced it in Institutiones calculi Differentialis (1755). The modern researches on the FDM started after the paper by Courant, et al. (1928), where the method was used to obtain approximated solutions to Partial Differential Equations (PDEs). In this field, the method was improved mainly after the Second World War when powerful computers were available. The books by Collatz (1966); Forsythe and Wasov (1960), and Richtmyer and Morton (1967) had a great role in stimulating research on the FDM. Other books by Cooper (1998); Kharab and Guenther (2002) considered Matlab applications too. Today, the FDM is considered a consolidated tool that is able to provide reliable solutions of PDEs and is used by scientists and technicians in many scientific areas, e.g., D’Acunto (2004); de Vahl Davis (1986). In this chapter, FDM will be applied to the heat equation by introducing these noteworthy methods: Explicit Euler Method Implicit Euler Method and Crank–Nicolson Method…
