Euler stability region

Figure 2.4 Stability region of the forward Euler method, linear equation to do so ...

The fourth-order explicit Runge-Kutta algorithm has a slightly better stability region than the Euler forward method. 

This defines the stability region for Euler s method as a disk in the complex hX-plane centered around —1 of radius 1. This region is shown in Fig. 4.1. If hX, where A is an eigenvalue of A lies in the indicated stability region, then Euler s method will be stable for the linear system z = Az. 

Fig. 4.1 Stability regions for Euler s method (l ) and Symplectic Euler/Verlet (right). When a harmonic oscillator is treated using these methods, the origin is unstable for Euler s method, regardless of stepsize—this means that there is no choice of scaling h which will allow us to ensure that 11 + ftA, < 1. On the other hand, the Verlet method has an interval of stability on the imaginary axis, and it is always possible to find a value of h which guarantees that hQ < 2...

In Fig. 4.8 the stability region of the three stage Radau Ila method is displayed. One realizes that the stability of the Radau method is much alike the stability of the implicit Euler method, though the three stage Radau method has order 5. Again we note the property i (0) = 1 which corresponds to zero stability in the multistep case. 

Figure 5.7 Stability region in the complex plane for Runge-Kutta methods of order 1 (explicit Euler), 2, 3, 4, and 5.

Figure 5.8 Stability region in the complex plane for the modified Euler (Euler predictor-corrector), Adams, and Adams-Moulton methods.

Thus, the imphcit Euler and Crank-Nicholson methods, or more generaUy (4.150) whenever 0 > 1/2, are A-stable. Here, we have only shown tiiis to be true when all stable eigenvalues are real. For four values of 9, Figure 4.11 plots the modulus of the growth coefficient tij = as a function of coj in the complex plane. The region of absolute stability,... 

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