Abstract
In this chapter, we discuss two time-stepping techniques that deliver second-order accuracy in time and, like the implicit Euler method, are unconditionally stable. One technique is based on the second-order backward differentiation formula (BDF2), and the other, called Crank–Nicolson, is based on the midpoint quadrature rule. Since the BDF2 method is a two-step scheme, it is not well suited to time step adaptation. Moreover, the stability analysis must account for the way the scheme is initialized at the first time step (we use here an implicit Euler step). In contrast to the BDF2 time stepping, the Crank–Nicolson scheme, like the implicit Euler scheme, is a one-step method. We will see however that the stability properties of the Crank–Nicolson method are not as strong as those of the implicit Euler method.
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