Maximum norm error estimates of the Crank–Nicolson scheme for solving a linear moving boundary problem

The Crank–Nicolson scheme is considered for solving a linear convection–diffusion equation with moving boundaries. The original problem is transformed into an equivalent system defined on a rectangular region by a linear transformation. Using energy techniques we show that the numerical solutions of the Crank–Nicolson scheme are unconditionally stable and convergent in the maximum norm. Numerical experiments are presented to support our theoretical results.