Variable time steps optimization of Lω -stable Crank–Nicolson method
Variable time steps optimization of Lω -stable Crank–Nicolson method
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Lω 稳定 Crank-Nicolson 方法的变时间步优化
DOI：
10.1515/1569398054308595
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
V. Lebedev
中科院分区：
文献类型：
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作者：
A. Medovikov;V. Lebedev
我们研究常微分方程的 Crank-Nicolson 方法（也称为欧拉二阶梯形法则 [5]）的优化。一阶常微分方程数值积分的 Crank-Nicolson 方法是 A 稳定的，但不是 L 稳定的。这意味着稳定区域与负半平面 z 完全重合：ℜz ≤ 0，但稳定函数 |R(z)|当 ℜz→ –∞ 时趋于 1 而不是 0。这会导致刚性微分方程数值解出现意外的振荡行为。为了避免这个问题，我们优化了稳定函数的稳定性。曲柄尼科尔森方法的步骤序列中的可变步骤使我们能够获得不同的稳定性函数，并针对稳定性函数的根和极点制定优化问题。该问题的最优解是经典有理Zolotarev函数。适当选择步长序列可以消除数值解的振荡行为。
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.