A. A. Medovikov, V. I. Lebedev
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.
Russian Journal of Numerical Analysis and Mathematical Modelling, Walter de Gruyter
Print ISSN: 0927-6467
Volume: 20, 06/2005
Pages: 283 - 303
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