4.5. A-stability

The plots of the region of absolute stability for the explicit RK4 method and the implicit Radau IA method are shown in Fig. 4.5 and Fig. 4.6. We can see that the region of absolute stability for the explicit method is bounded in the left-hand side of the complex plane, whereas the region of absolute stability for the implicit Radau IA method contains the entire left half-plane. This means that the step length used for the RK4 method needs to be less than some value when solving the test equation \(y = \lambda y\) with \(\text{Re}(\lambda) \leq 0\) in order for the method to be stable, whereas for the Radau IA method any step size may be used. This property is known as A-stability.

Fig. 4.5 Region of absolute stability for the RK4 method

Fig. 4.6 Region of absolute stability for the Radau IA method

Definition 4.9 (A-stability)

A method is A-stable if

\[ \mathbb{C}^- \subseteq S \]

where

\[ S = \{ z \in \mathbb{C} : |R(z)| \le 1 \}, \]

is the region of absolute stability, i.e., the left half of the complex plane lies entirely within the region of absolute stability.

Many stiff differential equations have eigenvalues with large negative real parts. For explicit methods, stability restrictions may force the use of extremely small step sizes. A-stable methods avoid this restriction because their stability regions contain the entire left half-plane.

Note

No explicit Runge-Kutta method can be A-stable [Dahlquist, 1963].

4.5.1. Testing for A-stability

Verifying A-stability from a plot can be difficult, so we require an algebraic test. Let

\[ R(z) = \frac{P(z)}{Q(z)}, \]

be the stability function of an implicit Runge-Kutta method, where \(P\) and \(Q\) have no common factors. A method is A-stable if

\[ |R(z)| \le 1 \qquad \forall \quad \text{Re}(z) \leq 0\]

To test whether a method is A-stable we need to check that

• there are no poles of \(R(z)\) in left half-plane

• on the boundary of the left half-plane \(|R(z)| \leq 1\).

The poles of \(R(z)\) are where \(Q(z) = 0\), so we compute the roots of \(Q\) and check whether the real parts are positive.

On the boundary of the left half-plane \(z = iy\) so

\[\begin{split} \begin{align*} \frac{|P(iy)|}{|Q(iy)|} & \leq 1 \\ |P(iy)| & \leq |Q(iy)| \\ 0 &\leq |Q(iy)| - |P(iy)|. \end{align*} \end{split}\]

Defining \(E(y)\) such that

\[ E(y) = |Q(iy)| - |P(iy)|, \]

then we need \(E(y) \ge 0\). To simplify the algebra we can square the complex modulus to eliminate the square root

\[ E(y) = |Q(iy)|^2 - |P(iy)|^2. \]

Checking that there are not poles in the left half-plane tells us that \(R(z)\) is analytic in this region. Checking that \(|R(z)| \le 1\) on the bounday of the left half-plane tells us that an analyic function cannot obtain a larger modulus in the interior than on the bounday. So by maximum modulus principle \(|R(z)| \le 1\) throughout the entire left half-plane.

Theorem 4.1 (Conditions for A-stability)

Let

\[ R(z) = \frac{P(z)}{Q(z)}, \]

be the stability function of an implicit Runge-Kutta method, where \(P\) and \(Q\) have no common factors. The method is A-stable if and only if the following are satisfied:

Condition A: All roots of \(Q(z)\) must have positive real parts

Condition B: The polynomial function

(4.9)\[ E(y) = |Q(iy)|^2 - |P(iy)|^2, \]

satisfies \(E(y) \ge 0\) for all \(y \in \mathbb{R}\).

Example 4.3

The Radau IA method is defined by the following Butcher tableau

\[\begin{split} \begin{array}{c|cc} 0 & \frac{1}{4} & -\frac{1}{4} \\ \frac{2}{3} & \frac{1}{4} & \frac{5}{12} \\ \hline & \frac{1}{4} & \frac{3}{4} \end{array} \end{split}\]

Determine whether this method is A-stable and plot the region of absolute stability.

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Solution

We saw in Example 4.2 that the stability function for the Radau IA method is

\[ \begin{align*} R(z) = \frac{1 + \frac{1}{3}z }{1 - \frac{2}{3}z + \frac{1}{6}z^2}. \end{align*} \]

Here \(Q(z) = 1 - \frac{2}{3}z + \frac{1}{6}z^2\) which has roots at \(z = 2 \pm \sqrt{2}\) which both have positive real parts so coniditon A is satisfied.

Using equation (4.9)

\[\begin{split} \begin{align*} E(y) &= |Q(iy)|^2 - |P(iy)|^2 \\ &= \left| 1 - \frac{2}{3}iy - \frac{1}{6}y^2 \right|^2 - \left| 1 + \frac{1}{3}iy\right|^2 \\ &= \left( 1 - \frac{1}{6}y^2 \right)^2 + \left( \frac{2}{3} y \right)^2 - 1^2 - \left( \frac{1}{3}y \right)^2 \\ &= 1 - \frac{1}{3}y^2 + \frac{1}{36}y^4 + \frac{4}{9}y^2 - 1 - \frac{1}{9}y^2 \\ &= \frac{1}{36}y^4 \geq 0 \qquad \forall y \in \mathbb{R}. \end{align*} \end{split}\]

So condition B is satisfied. Since both criterion A and B are satisfied then we can say that this is an A-stable method.