ESAM 446-1

Numerical Solution of Partial Differential Equations
Fall 2000
Hermann Riecke


Problem Set 2



This problem set is due October 16 in class.

  1. 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.

  2. 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.

  3. Use the result of problem 2 to find the second-order backward-difference approximation of du/dx. Do not derive it from scratch!

  4. Show that every central-difference approximation for du/dx vanishes at kh = p.

  5. Consider the initial-value problem
    t u = -x u,        u(x,0) = cos(2px),        0 £ x £ 10
    (1)
    with periodic boundary conditions.

    1. Find the exact solution.
    2. Compute the numerical solution for t = 2 using the following schemes and parameters1:

      1. Forward Euler with central differences, Dx = 0.01, and a sequence of Dt, Dt = 0.01, 0.005, 0.0025.
      2. Forward Euler with forward differences in space, Dx = 0.01 and a sequence of Dt, Dx = 0.01, 0.011, 0.005.
      3. Forward Euler with backward differences in space, Dx = 0.01 and a sequence of Dt, Dx = 0.01, 0.011, 0.005.
      4. 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.