Implicit and explicit Euler schemes

Abstract

The goal is now to discretize in time the space semi-discrete parabolic problem considered in the previous chapter. Since this problem is a system of coupled (linear) ODEs, its time discretization can be done by using one of the numerous time-stepping techniques available from the literature. In this chapter, we focus on the implicit (or backward) Euler scheme and on the explicit (or forward) Euler scheme, which are both first-order accurate in time. Second-order implicit schemes called BDF2 and Crank–Nicolson are investigated in the next chapter. The standard viewpoint in the literature is to interpret the above schemes as finite differences in time. This is the perspective we adopt in this chapter and the next one. We broaden the perspective later on by introducing a discrete space-time formulation and by considering higher-order time discretization methods.