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lostinphysics44
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Trying to understand what is going on here, and how this technique removes the dependence of sample rate from the solution.
Would it be possible to plot the radius change over time in matlab? How would I go about doing that?RUber said:If you Taylor expand out the function in time, you get for time step h:
##r(t+h) = r(t) + h r'(t) + \frac{h^2}{2} r''(t) + \frac{h^3}{3!} r'''(t)...\\
r(t - h) = r(t) - h r'(t) + \frac{h^2}{2} r''(t) -\frac{h^3}{3!} r'''(t)... \\
r(t-2h) = r(t) - 2h r'(t) + \frac{4h^2}{2} r''(t) -\frac{8h^3}{3!} r'''(t)... ##
##3r(t) - 3r(t-h) + r(t-2h) = r(t) +h r'(t) + \frac{h^2}{2} r''(t) -\frac{5h^3}{6} r'''(t)... ##
So you can see this scheme is accurate to ##\mathcal{O}(h^3). ##
The benefit of this method is that it looks like an semi-implicit method, which should become stable regardless the sample rate. The accuracy will still be dependent on sample rate, but the stability is much improved compared to forward difference methods.
RUber said:In the attachment, the equation marked (6.46) says
##\frac{dr_1}{dt} = f(\hat r _1, t_1).##
Do you know that function? If not, you can approximate the derivatives simply, but you lose some accuracy.
In order for this to work I have to assume a starting bubble radius yes? This is probably not the right forum for this, but think if i took a shot at a MATLAB script you could look it over for me?RUber said:That seems to work, remember that you are using the approximation ##\hat r_1 ## to approximate the derivative, so plug that estimate (based on prior r data) into the formula for dr/dt.
Once you have that estimate for dr/dt, you apply the averaging and approximation in 6.48 and 6.49.
Assuming you know the previous time data ##r_{i-1},r_{i-2},r_{i-3}##:
1. Set dt timestep.
2. Set pressure rate.
3. Calculate approximation ##\hat r_i## based on previous r information.
4. Use formula with pressure and ##\hat r_i## or ## r_{i-1}## to get ##\frac{dr_i}{dt}## and ##\frac{dr_{i-1}}{dt}##
5. Use equation 6.48 to get average rate of change for the timestep.
6. Use equation 6.49 to approximate ##r_i##.
RUber said:Based on the example, it looks like you would want to have the radius at the first 3 times. You can surely assume that r = 0 or r = r0 for times prior to start.
Yes, I can look at a script for you.
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