Variable time steps optimization of Lω -stable Crank–Nicolson method

We study the optimization of the Crank–Nicolson method, also known as the Euler second-order trapezoidal rule [5] for ordinary differential equations. The Crank–Nicolson method for the numerical integration of the first-order ordinary differential equations is A-stable, but it is not L-stable. This implies that the stability region coincides exactly with the negative half-plane z : ℜz ≤ 0, but the stability function |R(z)| tends to 1 rather than zero as ℜz→ –∞. This causes the unexpected oscillatory behaviour of the numerical solution of stiff differential equations. In order to avoid this problem we optimize the stability property of the stability function. Variable steps within the sequence of steps by the Crank–Nicolson method allow us to obtain different stability functions and formulate an optimization problem for roots and poles of the stability function. The optimal solution of this problem is the classical rational Zolotarev function. The appropriate selection of the sequence of step sizes eliminates the oscillatory behaviour of the numerical solution.