On iterated Crank–Nicolson methods for hyperbolic and parabolic equations

Different versions of the iterated Crank–Nicolson method are considered and their relation to the explicit Runge–Kutta methods is studied. Stability and accuracy properties of these methods are compared for both hyperbolic and parabolic equations. It is shown that Runge–Kutta methods offer more accurate and stable options even for a few evaluations of the right-hand sides. When applied to nonlinear equations, the iterated Crank–Nicolson methods have an accuracy barrier, which does not appear for Runge–Kutta methods.