Consider a crude model for a sphere in free fall, in which density, area, and drag coefficient are all assumed constant,
D(u) = ρu|u|CDA.
Here u|u| replaces u2 so that the direction of drag is always opposite that of the velocity. Assuming the only forces are drag and gravity, the downward velocity u obeys the ODE
du ρ
= f(u) = g − u|u|, (1) dt 2β
where β = CmDA is the ballistic coefficient.
(b) Linearize the ODE about the point u = 0. What type of solutions does the resulting linear ODE have? What does this model represent physically?
(c) Determine the terminal velocity, u∗, in terms of ρ, g, and β. You should not need to do any simulations or ODE integrations.
(d) Linearize the ODE about the point u = u∗. Express your answer as an ODE with dependent variable ˜u, where u = u∗ + u˜. Describe the solution of the linearized equation starting from the intitial condition u(0) = 0.
1
Problem 2: Integrating ODEs
We consider two explicit numerical integration schemes,
vn+1 − vn = ∆tf(vn) (Forward Euler) (2)
vn+1 − vn = ∆t 3f(vn) − f(vn−1) (Adams-Bashford) (3)
(a) Detemine if the above scehems are consistent, zero stable and convergent. If they are convergent, determine the order of accuracy.
(b) Calculate the exact solution to 1 in the previous problem assuming that u ≥ 0 for all t, and u(0) = 0. Assume that the equation is nondimensionalized in such a way that g = 10 and β = ρ/2.
(c) Write a Matlab code to compute the numerical solution for 0 < t ≤ T = 2 and the following five stepsizes ∆t ∈ {10−1,10−1/2,10−1/22,10−1/23,10−1/24} using methods 2 and 3. For each scheme, show the solutions for the different timesteps on the same plot. Note: For the first sime step of method 3, you can use method 2 since v−1 is not avalaible.
(d) Record the global error
max |vn − u(n∆t)| n=[0,T/∆t]
for each choice of ∆t. Plot the global error vs. the time step on a log-log plot for both schemes on the same plot and comment on the observed slope and its relationship to the global order of accuracy.
(e) Using the integration scheme 3 and ∆t = 10−1/24, solve the the nonlinear equation 1 and the linearized equation derived in Problem 1 (d) for the initial condition u(0) = −2 (initial upwards velocity) and 0 < t ≥ T = 3. Use the same nondimensionalization as in part (b). Show both solutions on the same plot and comment on the results.
2
