## APMA2821N - Homework

All page numbers refer to text.

1. Due on
February 15, 2012
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
February 22, 2012
1. 3.1 (p. 67)
2. 3.3 (p. 68) -- consider forward Euler and 2nd order methods to solve the problem. Determine the maximum stable timestep in all 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 February 29, 2012
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 March 7, 2012
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 propose an ERK with this property, B) Optimize the extension of the stability region along the negative real axis and propose 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 March 14, 2012
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 implement 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.
1. 4.10 (p.114). Use your own embedded ERK method. NOTE: Misprint: The parameter D should be D=90.5*0.4184e-3.
2. 4.12 (p.115). Use the 3rd order SDIRK (p.106) to solve the problem.

6. Due on
April 4, 2012

7. Due on
April 11, 2012
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
April 18, 2012
1. 5.4 (p. 154) - enough to only consider orders 1-6 (table 5.3)
2. 5.5 (p. 154) - use your error controlled Dormand-Prince 4(5) scheme to solve the problem
3. 5.5 (p. 154) - implement a 3rd order predictor-corrector AB with the Milne device to enable adaptivity. Compare with the results obtained using the ERK in 2).
4. 5.11 (p. 157)

9. Due on
April 25, 2012