Partial differential equations and other higher-dimensional problems

Questions
Q6.1 Discuss the finite-difference time-domain method.
Q6.2 What are some of the commonly encountered PDEs? What methods are
used to solve them?
Q6.3 Discuss the scaling of Monte Carlo methods for high-dimensional integration.
Q6.4 Why would we use a Lorentzian distribution rather than a Gaussian distribution
for Monte Carlo integration when we don’t know an upper bound
for the function?
Q6.5 What are other commonly used Monte Carlo methods?
Q6.6 What other applications are there for random numbers?
Exercises—due Friday 31st October
Required
The grade of 1–5 awarded for these exercises is based on the required exercises. If all of
the required exercises are completed correctly, a grade of 5 will be obtained.
R6.1 Q1 in Sauer computer problems 8.1, 8.2, or 8.3. (One of these will do. If you
want to do more, see additional exercises below!)
a) From Sauer 8.1: Solve ut = 2uxx for 0 ≤ x ≤ 1, 0 ≤ t ≤ 1, for the sets
of boundary conditions
i. u(x, 0) = 2 cosh x for 0 ≤ x ≤ 1
u(0, t) = 2 exp(2t) for 0 ≤ t ≤ 1
u(1, t) = (exp(2) + 1) exp(2t − 1) for 0 ≤ t ≤ 1
ii. u(x, 0) = exp x for 0 ≤ x ≤ 1
u(0, t) = exp(2t) for 0 ≤ t ≤ 1
u(1, t) = exp(2t + 1) for 0 ≤ t ≤ 1
MODULE 6. 65
using the forward difference method for step sizes h = 0.1 and
k = 0.002. Plot the approximate solution (the mesh command might
be useful). What happens if you use k > 0.003? Compare with the
exact solutions if possible.
b) From Sauer 8.2: Solve the following initial–boundary value problems
using the finite difference method with h = 0.05 and k = h/c. Plot the
solutions.
i. utt = 16uxx
u(x, 0) = sin(πx) for 0 ≤ x ≤ 1
ut(x, 0) = 0 for 0 ≤ x ≤ 1
u(0, t) = 0 for 0 ≤ t ≤ 1
u(1, t) = 0 for 0 ≤ t ≤ 1
(Solution is u(x, t) = sin(πx) cos(4πt))
ii. utt = 4uxx
u(x, 0) = exp(−x) for 0 ≤ x ≤ 1
ut(x, 0) = −2 exp(−x) for 0 ≤ x ≤ 1
u(0, t) = exp(−2t) for 0 ≤ t ≤ 1
u(1, t) = exp(−1 − 2t) for 0 ≤ t ≤ 1
(Solution is u(x, t) = exp(−x − 2t))
c) FromSauer 8.1: Solve the Laplace equation for the following boundary
conditions using the finite difference method with h = k = 0.1. Plot
the solutions.
i. u(x, 0) = sin(πx) for 0 ≤ x ≤ 1
u(x, 1) = exp(−π) sin(πx) for 0 ≤ x ≤ 1
u(0, y) = 0 for 0 ≤ y ≤ 1
u(1, y) = 0 for 0 ≤ y ≤ 1
(Solution is u(x, y) = exp(−π y) sin(πx))
ii. u(x, 0) = 0 for 0 ≤ x ≤ 1
u(x, 1) = 0 for 0 ≤ x ≤ 1
u(0, y) = 0 for 0 ≤ y ≤ 1
u(1, y) = sinh π sin(π y) for 0 ≤ y ≤ 1
(Solution is u(x, y) = sinh(πx) sin(π y))
R6.2 Use a Monte Carlo method to find the area of a circle, the volume of a
sphere, and so on, for higher-dimensional “circles” and “spheres”. Make a
statistical estimate of the accuracy of your result, and compare with known
results.
Additional
Attempts at these exercises can earn additional marks, but will not count towards the
grade of 1–5 for the exercises. Completing all of these exercises does not mean that 4
marks will be obtained—the marks depend on the quality of the answers. It is possible to
earn all 4 marks without completing all of these additional exercises.
A6.1 Selections from Sauer computer problems 8.1, 8.2, 8.3.
A6.2 Find a PDE solver. Discuss the code, if source code is available. Test, and
discuss.
A6.3 Find or implement code for high-dimensional optmisation. Test, and discuss.
66 COSC2500/COSC7500—Semester 2, 2014
A6.4 Implement Monte Carlo integration for circles, spheres, etc., using C or
some other compiled language. Compare the performance of your code
with your Matlab version. Investigate the effect of using single precision
instead of double precision.

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