Jeffrey Hellrung
Friday, November 09, 2005
Math 269A, Assignment 06
1. Consider the two stage Runge-Kutta method
y∗=yn+αdtf(yn),
yn+1 =yn+dt (β1f(yn) + β2f(y∗)) .
Assume that the constants α,β1, and β2are chosen so that the method is second order (Assignment
3, Problem 1).
(a) Find the interval of absolute stability for the method.
(b) Does the interval of absolute stability depend upon the coefficients α,β1, and β2?
Solution
From Assignment 3, Problem 1, we have that
β1+β2= 1, αβ2=1
2.
(a) Setting f(y) = λy, we obtain
y∗=yn+αhf(yn) = yn+αλhyn,
yn+1 =yn+h(β1f(yn) + β2f(y∗))
=yn+h¡β1λyn+β2λyn+αβ2hλ2yn¢
=¡1 + (β1+β2)λh +αβ2(λh)2¢yn
=¡1 + λh + (λh)2/2¢yn
,
hence we require that
¯¯¯¯
1 + λh +(λh)2
2¯¯¯¯
<1⇒λh ∈(−2,0).
(b) Clearly, the interval of absolute stability is independent of α,β1, and β2(assuming these constants
are chosen such that the method is second order).
2. Consider the ODE whose numerical solution you computed in Assignment 4, Computational Problem
3. dy
dt =µ0 1
−β−α¶y, y(0) = µ1
0¶.
(a) When α= 100.0 and β= 1.0 determine the eigenvalues of the Jacobian of this system of equations.
(b) Estimate the time-step restriction so that
i. Forward Euler
ii. 4th Order Runge-Kutta
yield a qualitatively correct solution. This is just an estimate, so you can do it graphically.
(Attached are the plots of the regions of absolute stability for Runge-Kutta methods of orders
1-4.)
(c) Use your program from Assignment 4 and check the validity of your estimate. For each method,
determine if your timestep estimate is close to the timestep required to get a “reasonable looking”
numerical solution.
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