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MCEN 3030: Practice exam for final (100pts)
You can use your notes, books, calculators, computers but not a colleague...
1. ODE: Runge-Kutta methods
A) Consider the following Butcher tableaux and classify them (ERK,DIRK,IRK)
B) For both methods show what is the order of the method (local truncation error) show all your calculations.
2. ODE: Linear-multistep methods (LMM)
A) There is something wrong with this LMM:
yn+2 = − 4 yn+1 + 5yn + h(4f (xn+1, yn+1) + 2f (xn, yn))
Find out and show what is the problem.
B) Consider the BDF method
yn+1 = yn− 1 + 2hf (xn, yn)
which is obtained by using a centered difference formula on dydx |n.
- Find the order of this method (or local truncation error).
- Show that the method is 0-stable.
- Rewrite the method as a one step method using zk = (yk, yk− 1 )
3. Iterative methods for linear systems
Consider the following linear system: [ 2 − 1 − 1 2
] [
u 1 u 2
]
[
]
A) Show that A is SPD.
B) Split A into L + D + U and apply 2 Jacobi iterations using u(0)^ = (0, 0)T^ as a starting vector.
C) Apply 2 iterations of the CG algorithm using u(0)^ = (0, 0)T^ as a starting vector.
4. Parabolic PDEs
For the PDE ∂u ∂t
∂^2 u ∂x^2 discretize using implicit Euler in time and centered finite difference in space. A) Find the order of this method in space and time(Taylor ...).
B) Study the stability of the overall problem.
C) If you take a very big ∆t (because you can) why will this not be a very good approach to solve your problem?
D) From A and B what can you infer about the global convergence of this method?
NOTES:
Runge-Kutta (s stages solving dydx = f (x, y(x))):
Yi = yn + h
∑^ s
j=
aij kj ,
ki = f (xn + cih, Yi),
yn+1 = yn + h
∑^ s
i=
biki (1)
Linear-Multistep methods (k steps)
∑^ k
i=
αiyn+i = h
∑^ k
i=
βif (xn+i, yn+i) (2)
Useful lemma: For all matrix A ∈ Rn×n^ and any > 0 exists a norm || · || such that ρ(A) < ||A|| < ρ(A) + holds. (ρ(A) = max 1 ≤i≤n |λi| where the λi are the eigenvalues of A).
