Zero stability

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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 ${\displaystyle y'(x)=0}$ .

A linear multistep method is zero-stable if all roots of the characteristic equation that arises on applying the method to ${\displaystyle y'(x)=0}$ 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#edit

The following third-order method has the highest order possible for any explicit two-step method^[2] for solving ${\displaystyle y'(x)=f(x)}$ : ${\displaystyle y_{n+2}+4y_{n+1}-5y_{n}=h(4f_{n+1}+2f_{n}).}$ If ${\displaystyle f(x)=0}$ identically, this gives a linear recurrence relation with characteristic equation ${\displaystyle r^{2}+4r-5=(r-1)(r+5)=0.}$ The roots of this equation are ${\displaystyle r=1}$ and ${\displaystyle r=-5}$ and so the general solution to the recurrence relation is ${\displaystyle y_{n}=c_{1}\cdot 1^{n}+c_{2}(-5)^{n$ . Rounding errors in the computation of ${\displaystyle y_{1$ would mean a nonzero (though small) value of ${\displaystyle c_{2$ so that eventually the parasitic solution ${\displaystyle (-5)^{n$ would dominate. Therefore, this method is not zero-stable.

References#edit

^ 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. JSTOR 24490010.
^ a b 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.

Categories Numerical differential equations

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