FINAL
Problem 1
To demonstrate L-stability concept we consider x0 = λ(x − cos(t)) − sin(t), λ = −106 ,
x(0) = x0 . The exact solution of this equation is given by
x(t) = eλt (x0 − 1) + cos(t).
Use backward Euler and trapezoidal rule (2nd order Adams-Moulton, xn+1 − xn =
0.5h(fn+1 + fn )) to solve numerically this equation with x0 = 1 and x0 = 1.5. In all cases
use h = 0.1, N = 400. Compare the solutions with the exact one (by plotting).
Note that both trapezoidal and backward Euler methods are A-stable, while only Backward Euler method is L-stable
Problem 2
Consider the multi-step method (2nd order BDF method)
3xn+2 − 4xn+1 + xn = 2hfn+2 .
(1)
a) Find the region of absolute stability of the method (note that the method is A-stable).
b) Show that the method is L-stable.
c) Note that explicit methods can not be L-stable. To show this consider an “explicit
version” of (1)
3xn+2 − 4xn+1 + xn = 2hfn+1 .
Though for this explicit method q(z) = z satisfies root condition show that the method is
not L-stable.
d) (bonus) Why explicit multi-step methods can not be L-stable.
Problem 3
Consider A-stability of one-step methods (e.g., Runge-Kutta methods).
a) Write down in terms of ω (do not plot) A-stability for 2nd and 4th order Runge-Kutta
methods. (note you only need to write down A-stability regions as W (ω) = 0).
b) How can you adopt the idea of boundary locus method to plot A-stability region for
2nd order Runge-Kutta method? Plot the A-stability region.
Problem 4
Linear multi-step methods can be considered as one step methods. Assume ak = 1,
bk = 0 (without loss of generality) and x0 = λx. Show that the multi-step method
ak xn + ...a0 xn−k = h(bk fn + ... + b0 fn−k )
(2)
un = Aun−1 ,
(3)
can be written as
where un = (xn−k+1 , ..., xn )T with some matrix A. Find A.
Show that A-stability of (3) is the same as the A-stability we have derived for (2).
Note. If you do not want to work for arbitrary k you can assume k = 4.
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Problem 5
Given
xn + αxn−1 + (−1 − α)xn−2 = h(βfn + (2 + α − β)fn−1 ).
a) Determine the relationship between α, β such that the method is consistent, convergent
with second order accuracy. Find all possible values of α for which the method is (zero)
stable.
b) Find a value for α (or β) such that the method is absolutely stable.
c) Is the method with calculated values of α and β L-stable?
Problem 6
Consider a sequence of polynomials φn (x) (degree n) that is w-orthogonal to q(x) of lesser
degree on (a, b), i.e.,
Z b
w(x)φn (x)q(x)dx = 0.
a
a) Show that φn (x) can be written as
φn (x) =
1 dn
Un (x),
w(x) dxn
where Un (x) is a solution of
dn+1 1 dn
Un (x) = 0,
dxn+1 w(x) dxn
(4)
and Un (x) and its derivatives up to n − 1 order vanishes at x = a and x = b.
Note that the solution of (4) exists and unique up to a constant multiplier.
b) Consider quadrature formula
Z
∞
−x
e f (x)dx ≈
0
n
X
Ai f (xi ).
(5)
i=1
Show that if w(x) = e−x , then Un (x) = e−x xn satisfy (4). Using the latter write down
formula for w-orthogonal polynomials φn (x) for n = 3.
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