APMA2811F - Homework


All page numbers refer to text.

1. Due on September 30, 2009
  1. 2.1 (p. 34)
  2. 2.2 (p. 34)
  3. 3.3 (p. 68) -- use a forward Euler method to solve the problem.

2. Due on October 7, 2009
  1. 3.1 (p. 67)
  2. 3.3 (p. 68) -- consider forward Euler, 2nd order and 4th order Runge-Kutta methods to solve the problem. Determine the maximum stable timestep in all three cases and confirm the analysis with computational evidence.
  3. 3.3 (p. 68) -- consider backward Euler and Crank-Nicholson methods to solve the problem. Confirm the unconditional stability of the two methods.
  4. 3.9 (p. 70)

3. Due on October 21, 2009
  1. 3.4 (p. 68)
  2. 4.1 (p. 111)
  3. 4.2 (p. 111)
  4. 4.8 (p. 113)
  5. 4.12 (p. 115)

4. Due on October 28, 2009
  1. 4.6 (p. 112)
  2. 4.19 (p. 121)
  3. Use rooted trees to derive the order conditions for a 5th order Runge-Kutta methods.
  4. Consider a 2nd order, 3 stage explicit Runge-Kutta method. Use the additional freedom to A) Optimize the extension of the stability region along the imaginary axis and proposed an ERK with this property, B) Optimize the extension of the stability region along the negative real axis and proposed an ERK with this property. C) Test the accuracy and stability of the two methods to validate the analysis. You can choose your own simple test example.

5. Due on November 4, 2009
  1. Consider the Dormand-Prince 4(5) pair (p.93). A) Derive and plot the stability regions for both ERK in the method. B) Discuss and implemennt an adaptive step control using the Dormand-Prince method. C) Use this method to solve the astronomy problem in 4.12 (p.115) and try to estimate how many steps you need to obtain a visually correct solution. Compare with the results you obtained in Homework 3.
  2. 4.10 (p.114). Use your own embedded ERK method. NOTE: Misprint: The parameter D should be D=90.5*0.4184e-3.
  3. 4.12 (p.115). Use the 3rd order SDIRK (p.106) to solve the problem.

6. Due on November 11, 2009
  1. Problem description
  2. IMEX-RK Tableau
  3. CUSP test case

7. Due on November 18, 2009
  1. 5.2 (p. 153) - use the unstable method in Example 5.6 in a).
  2. 5.9 (p. 156) - Use an adaptive ERK in b) and suitable BDF method in c).
  3. 5.10 (p. 157)
  4. 5.12 (p. 157)

8. Due on December 2, 2009
  1. Problem set
  2. Multibody problem description