Zero stability
Zero-stability, also known as D-stability in honor of Germund Dahlquist,[1] refers to the stability of a numerical scheme applied to the simple initial value problem [math]\displaystyle{ y'(x) = 0 }[/math].
A linear multistep method is zero-stable if all roots of the characteristic equation that arises on applying the method to [math]\displaystyle{ y'(x) = 0 }[/math] have magnitude less than or equal to unity, and that all roots with unit magnitude are simple.[2] This is called the root condition[3] and means that the parasitic solutions of the recurrence relation will not grow exponentially.
Example
The following third-order method has the highest order possible for any explicit two-step method[2] for solving [math]\displaystyle{ y'(x) = f(x) }[/math]: [math]\displaystyle{ y_{n+2} + 4 y_{n+1} - 5y_n = h(4f_{n+1} + 2 f_n). }[/math] If [math]\displaystyle{ f(x)=0 }[/math] identically, this gives a linear recurrence relation with characteristic equation [math]\displaystyle{ r^2 + 4r - 5=(r-1)(r+5) = 0. }[/math] The roots of this equation are [math]\displaystyle{ r=1 }[/math] and [math]\displaystyle{ r=-5 }[/math] and so the general solution to the recurrence relation is [math]\displaystyle{ y_n = c_1\cdot 1^n + c_2 (-5)^n }[/math]. Rounding errors in the computation of [math]\displaystyle{ y_1 }[/math] would mean a nonzero (though small) value of [math]\displaystyle{ c_2 }[/math] so that eventually the parasitic solution [math]\displaystyle{ (-5)^n }[/math] would dominate. Therefore, this method is not zero-stable.
References
- ↑ Dahlquist, Germund (1956). "Convergence and stability in the numerical integration of ordinary differential equations". Mathematica Scandinavica 4 (4): 33–53. doi:10.7146/math.scand.a-10454. https://www.jstor.org/stable/24490010. Retrieved 19 July 2022.
- ↑ 2.0 2.1 Hairer, Ernst; Nørsett, Syvert; Wanner, Gerhard (1987). Solving Ordinary Differential Equations I. Berlin: Springer-Verlag. pp. 326–328.
- ↑ Butcher, John C (1987). The Numerical Analysis of Ordinary Differential Equations. Wiley. p. 11.
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