(Note: This homework will not be collected, and will not be graded. Think of this as a Practise Midterm (midterm will be shorter, but this is the difficulty you should expect!)
Problem 1
show that the Region of Absolute Stability for the trapezoidal method is (-infinity,0).
Problem 2:
Consider an initial value problem of the form
dy/dt = F(y)
y(t0) = y0
This is called an autonomous differential equation, because F(y) does not depend explicitly on t. Give an expression
for the leading term of the truncation error for the Trapezoidal method
Problem 3:
Consider the initial value problem
dy/dt = t*sin(y)*cos(2*y)
0 <= t <= 2.51
y(0) = 1
Estimate a restriction on the timestep you can use applying a Runge Kutta scheme of order 3, such that the solution
exhibits qualitatively the correct behavior.
The region of absolute stability for a third order Runge Kutta scheme is (-2.51,0).
Problem 4:
Consider applying the Trapezoidal method
wk+1 = wk + h/2*f( wk , tk) + h/2*f(wk+1 tk+1)
To the differential equation dy/dt = 4t2cos(y).
(a) Write out the equation that must be solved at each timestep to advance the solution from wk to wk+1.
(b) Write out the Newton's method iteration that could be used to solve the equation in (a).
(c) Newton's method requires an initial guess; what would you choose as an initial guess for the iteration in (b)
?
Problem 5:
Consider the third order differential equation
d3y/dt3 - 3 dy/dt = 6
y(0) = 1
dy/dt(0) = 2
d2y/dt2(0) = 3
(a) Put this equation into an equivalent first order system form.
(b) Express the system in (a) using matrix/vector notation i.e. give an expression for du/dt if your unknowns
are the components of a vector u.
Problem 6:
Assume that the numerical approximation of a certain method produces the results given below at a time t=2.0. The
methods was used with different timesteps, as indicated. Can you estimate the order of the method ? (Hint: Assume
that the result for the small timestep h=0.0001 "is almost exact").
|
h |
Approximation |
|
0.0001 |
1.00000 |
|
.1 |
0.999439 |
|
.05 |
0.999849 |
|
.025 |
0.9999605 |
|
.0125 |
0.99999 |