ESAM 446-1
Numerical Solution of Partial Differential Equations |
This problem set is due October 16 in class.
- Derive the 6th order central difference approximation of d2u/dx2,
i.e. the error will be of O(Dx6) using i) Taylor expansion of u(x+h)
and ii) Fourier analysis.
Find the Fourier representation of the operator obtained in this approximation and
the leading-order term in the relative error.
- Derive the second-order approximation of du/dx
using forward differences, i.e. involving only points to the right of the point
at which the derivative is to
be approximated. Do a Fourier analysis of this approximation and find again the
leading-order term in the relative error (in real and in imaginary part). Discuss the
effect of the two terms if this scheme is used in the PDE ¶t u = ¶x u.
- Use the result of problem 2 to find the second-order
backward-difference approximation of du/dx. Do not derive it from scratch!
- Show that every central-difference approximation for du/dx vanishes at kh = p.
- Consider the initial-value problem
¶t u = -¶x u, u(x,0) = cos(2px), 0 £ x £ 10 |
| (1) |
|
with periodic boundary conditions.
- Find the exact solution.
- Compute the numerical solution for t = 2
using the following schemes and parameters1:
- Forward Euler with central differences, Dx = 0.01, and
a sequence of Dt, Dt = 0.01, 0.005, 0.0025.
- Forward Euler with forward differences in space, Dx = 0.01 and
a sequence of Dt, Dx = 0.01, 0.011, 0.005.
- Forward Euler with backward differences in space, Dx = 0.01 and
a sequence of Dt, Dx = 0.01, 0.011, 0.005.
- Lax-Friedrich with Dx = 0.01 and
a sequence of Dt, Dx = 0.01, 0.013, 0.005.
For each scheme describe the results in a few words and show one representative
plot of the solution. How do the results
compare to your expectations based on Neumann analysis?
Footnotes:
1Note:
You can obtain a matlab template for a program that also shows
a movie of the simulation on the class web site.
File translated from
TEX
by
TTH,
version 2.60.
On 5 Oct 2000, 07:22.