Numerics

Question 1

Difference between multi-step and multi-stage methods

Answer 1

Multi-stage: doesn’t use any previously computed values

Multi-step: evaluates F at existing time steps

Question 2

Which method is AM method identical to?

Answer 2

Trapezoidal Implicit method

Question 3

Difference between IVP and BVP

Answer 3

IVP: solution is constrained by initial conditions
BVP: solution is constrained by boundary conditions

Question 4

Total operations/steps of Guassian and LU

NxN matrix

Answer 4

G: O(N^3)
LU: O(N^2)

Question 5

Q(s)>0

Answer 5

f increases in all directions away from minimum
global minimum
convex

Question 6

Positive definite matrix

Answer 6

all eigenvalues > 0

Question 7

H symmetric implies..

think Eigen

Answer 7

its eigenvalues are real and its eigenvectors are orthogonal

Question 8

What is the Leapfrog scheme derived from?

Answer 8

The centred difference formula

Question 9

What method is the AM2 identical to?

Answer 9

Trapezoidal Implicit Method

Question 10

Dirichlet BCs

Answer 10

y(x_j)=a_j j=1 and/or 2

Question 11

Neumann BCs

Answer 11

y’(x_j)=b_j j=1 and/or 2

Question 12

Common choice a,b,c,d for RK2

Answer 12

c = d =1/2
a = b = 1

Question 13

Choice of a,b,c,d for RK2 Heun’s scheme/improved Euler method

Answer 13

c = 0
d = 1;
a = b =1/2

Question 14

RK2 constraints on a,b,c,d

Answer 14

c+d=1

da=db=1/2