 Right now what we have done is in the last class we saw demonstration of package called maxima that I use to derive basically the modified equation okay and amongst various things so we saw we have already seen the effect that the extra terms that you get in the modified equation. We have investigated in earlier classes what is the effect of adding the second derivative term, what is the effect of adding the third derivative term, fourth derivative term and so on. And in the last class we actually saw that by adding appropriate terms to the differential equation it is possible for us in a targeted way to eliminate certain terms in the modified equation okay. So now what we will do is we will see, we will look at a demo right solving only the linear first order one dimensional wave equation okay, we will look at a demo. I started a little of it in the last class but we will just go or we will look at the demo and see whether all the properties that we have predicted, everything that we have predicted so far. See as opposed to Laplace's equation, Laplace's equation we just solved right, you had to went ahead and you solved it, it is a relatively what should I say robust problem right, I will explain to you maybe towards the end of the semester when we do a little calculus of variations where that idea comes from, I have already given you the essence of that but it is a relatively robust problem so I had no qualms about asking you to just go ahead and solve it numerically, you understand what I am saying writing a program. But by the time we came to a wave equation we did a little analysis and the idea was that you know we will find out whether it works, whether the schemes work and so on. There are issues, stability related issues right and there could be what do you call it the fact that the equation is modified and so on. And based on these things we have made predictions as to how the code will work and now we will actually run the code and see whether those predictions how they work out, to what extent are they true, how do we interpret these, how do we interpret the analysis that we have done, am I making sense okay right. So let me run that, this is by the way as you can see it is a program written in Python, I am using the user interface is built using a package called Python card which is built on WX windows. So what we will do is let me just start that off okay I quickly get rid of this, get rid of that. I just, I have made a few small changes since the last, the code is an evolving code, few small changes since the last time, for example if I say reset why I eliminated the reset anyway okay fine. So right now if it does not, if you cannot make out it is right now set, so this window that pops up here allows me to choose various and sundry schemes. It is set by default at FTCS, there is an FTCS 2, a variation that we will look at later which is basically FTCS with the second derivative term knocked out okay. So that according to us should have been a stabilizing influence, we will check that out. There is FTFS, FTBS so all the schemes that we have looked at, there is a Laxventer scheme, McCormack scheme which we have not looked at, you can go check that out and there is FTCS applied to heat equation. So all of it is wave equation except for the last one which is heat equation. So these are the schemes that I proposed to look at in this demo. We have already seen what happens when you apply FTCS to this equation. So basically it looks like you know it is supposed to be unstable and indeed it does behave as though it is unstable right, it is diverging. I have not bothered to rescale simply because we just want to know whether it diverges, once it goes beyond out of the screen right, it does not make any sense anymore. We can try increasing the number of grid points to see if that makes a difference. All that does is that that ramp looks a little better but wherever the ramp is it starts to oscillate and remember the notion of high frequency is associated with the grid size. So we can see that it is the high frequencies that are diverging right. So that this is near very close to the highest frequencies that are that can be represented on this. There is still propagation, the basic feature of the equation is still there, it is still propagating in the x direction, it is just that it is eventually going to diverge okay. So since I do not want to get into trouble with that, the other possibility that we thought about was would it work if I lowered my CFL, so instead of 1 if I made my CFL 0.1 would it work. So this is going to 0.1, so this is a little arithmetic that we have to get right, 0.1 given that the speed is the same and the grid size is the same. Lowering the CFL by 0.1 to 0.1 means that the delta t has become one-tenth right. So we really need to take 10 times the time step, I will increase the number of time steps, the little window here is the number of time steps, we really need to take 10 times the time steps to effectively get the same time step. And then you see that the behavior though it is smoother than last time, why is it smoother? It looks cleaner, it is not diverging as much, so the growth is not as fast right, the growth is not as fast because the mu2 term, the size of the mu2 term has gone down, the magnitude of the coefficient of the second derivative has gone down. But it still seems as though it is going to diverge right and there is nothing that we can do with this right, unless you want to try something smaller, if there is hope and sometimes this happens, we fall into this trap saying oh let me make it a little smaller maybe it will work for that right, always a little hope. So if you want to make it smaller since I made it 0.1 I will take 100 time steps at a time and we reset and who knows maybe it will actually work, nope there it goes. So it is not supposed to see the point is it is not supposed to exceed 1, this is like having a dam that suddenly breaks right, so I expect that the water to flow out and right now I mean you will see of course in the next class I will show you that it is possible that crests and peaks form in real life, it does happen right, but right now the simplistic model that we have we expect the water to just translate, we do not expect anything else and it does not look like that is going to happen right because of the, because I am taking I am doing more work, my program is taking more time because I am taking 100 time steps each time but we are not getting anything out of it, it is going to eventually diverge I mean it is there, all the symptoms are there, is that fine okay. So the next thing that we do is we look at FTFS, let us look at FTFS and go back to CFL1, I will reset it, what do we expect, I do not want 100 time steps maybe I will take this one time step, what do we expect, FTFS what do we expect, it is supposed to be unstable, you are going to blow up why do not I take 100 time steps, what is happening, I am actually clicking on that button right, I thought it is supposed to blow up, I have taken at least 400 time steps of it, the trouble is everything is 0, the only point that is non-zero is at the left boundary, am I making sense, the only point that is non-zero is at the left boundary and it is not participating in the computation, FTFS is only using points to the right of the first point and only the first point is non-zero, all the other points are 0, so the computation just is not starting off, there is nothing to do, am I making sense, you are just manipulating adding and subtracting 0s right, okay. So there may be times and it is possible that we look at this, you have to be very careful, so you have to the thing, one thing that you remember from this is you have to have a sense of what is the answer that you expect, especially when you are developing these codes you have to have a sense as to what is the answer that you expect, what is the behavior that you expect, okay. Yes, from the numeric point of view we expected it to diverge but from the physics point of view, the physics of the equation if I may say, I expected this wave to propagate, fact that it is not propagating right, tells us that there is an issue, the fact that it is not diverging does not mean that oh we are happy saying that oh I have a steady state solution, I managed to get the solution, it is very important, the fact that it is not diverging does not mean that we are and it is not changing, delta u is you understand the change in the solution from one time step to another time step is 0, so it is very easy to look at this equation and look at this and basically say oh I have got the solution I have a steady state solution, okay. So not true, let us try, let us try FTBS, FTBS of course I will take one time step and reset that just for good measure and as you would expect, right FTBS with sigma equals 1, right behaves the way we expected to behave, in fact if I reset it and I take, let me take 50 time steps, okay, it makes it half wave, I take 50 more time steps and it goes, it is actually just at the exit, it is just gone through the domain, am I making sense, okay, the propagation speed is right, I mean that it looks reasonable that the propagation speed is right, but of course the ramp is still there, so if you want to represent, if you want to represent right, if you want to represent that step, the closer you want to represent the step, the finer your grids will have to be, fine or you will have to do some fancy grid generation, okay, which is sort of outside the scope of this course, but you will have to do some fancy grid generation, okay, there are two ways we can go now, right, since FTFS we have already seen the stability condition did not quite work out the way it is supposed to work out, what I am planning to do is I am going to run CFL greater than 1, right, I do not know if you guys have tried this, I am going to run a CFL greater than 1, so I will run 1.1, 1.1 is supposed to diverge, 1.1 is supposed to diverge, okay, so I guess I could just take 50 time steps in one shot, so that on the other hand it may blow up, okay, fine, it take one time step, there is that peak, so it looks like it is going to go, right, it looks like it is going to go, they look sharper than, you notice they look sharper than what we got with FTCS, it looks a lot sharper than what we got with FTCS, okay. So is it worthwhile going you think or should I just stop, let me take 10 time steps at a time and see what happens, clearly it is diverging, but it is also being propagated out and it is gone, okay, so this is an important thing, so you can turn around and say wait a minute, it is supposed to be unstable, this program had no business working, well it depends on what you are looking for, if you are looking for the steady state solution, it actually works, okay, and why did it work because we are looking at a finite domain, I am looking at an interval here which goes from 0 to 2 pi, okay and now it is a matter of dynamics, it is blowing up, is it going out, like it is like getting to a game that ends in a tie in a sense, right, you do not know what is going to happen, you do not know which is going to, which is going to go, which is whether it is going to blow up or it is going to make it out, fine, okay. So you have a, you have a, you have a, you have a, if you are looking only for the steady state solution, the transient is clearly wrong, right, the transient is clearly wrong, fine, okay. So there are, there are, see these look, I know, I am sort of going through, there are some obvious, but there are some observations I want to make which are extremely important for the kind of work that you may do in CFD, okay. So if you are looking for the steady state solution, right now what I seem to have said is I do not care what the transient is like, as long as I get my steady state solution, okay. The second thing is your computational domain is finite, we cannot handle infinite computational domains, right. So unless you do something interesting, you cannot handle infinite computational domains, typically we truncate the domain, okay, typically we truncate the domain. And in that case, even if your scheme is unstable, it is possible that any unstable mode if it flows out of the system that you still get the steady state solution, fine, okay. So it is unstable, the scheme is unstable, 1.1 it should not have worked, maybe towards the end, right, I will come back to this and run, we will rerun this and maybe I will run 1.2 or 1.5 or whatever it is, see larger values. I do not want, I do not want to just crash my program right now, but, right, it is not so robust that I can recover from a crash, okay, fine. What about smaller values of CFL? What do you expect? If I run 0.5, any predictions? Did I do this in the last class? 0.5, if I take one time step, it looks the same, that looks a little different, you may not remember it. So maybe what I do is I take 0.5, I have halved the time step, right, so maybe I will take 20 time steps at a time, which is equivalent to what is happening, can you tell me what is happening? Now dispersion was what you saw when, not dispersion, what is this? FTBS, what did you expect? What was it when it was 1? It runs sigma equals 1 again. If you want to do a, just say instead of a ramp it is actually a, it is a step, okay, we know it is a ramp. So it is a ramp, you do a series expansion using Fourier series, right, in order to get the ramp, in order to get that ramp, all of those have to match exactly, they do not match exactly, you will start seeing oscillations, okay. So when we did 1.1, you actually saw that, so if you do 1.1, it starts to oscillate at that point, okay, it is not just diverging, there seems to be some, but if you do 0.5, maybe I will reset it, if you do 0.5, what is happening? What was the other phenomenon that we had? It is clearly stable, but there is something that even 1.0 is stable, just describe to me what has happened to the curve. The curve seems to have become smoother, in order to get that steep step, what do I need? See all the high frequency terms have disappeared, that is basically what is happened, okay, it is dissipated, decay, I mean I prefer the high frequencies are decaying, right, because at 0.5 basically what is happening is the high frequencies are going away, so the curve is getting smoothed out, right, the curve is getting smoothed out, am I making sense? So the high frequency terms are going away and the curve is getting smoothed out, if you are again looking only for the steady state, you do not care, but if your steady state has a discontinuity in it, then we have a problem, because this may not be what you want, right, this may not be what you want, it is supposed to be a step, instead of a step you get something that, so you could say that why do not we try a larger number of, why do not we try a larger number of grid points and see whether that makes a difference, so if I go to, I do not know, 201 I will just double it just for the fun of it, if I go to 201, yeah it is sharper but you can see that it is still getting rounded off, right, it is just that I can represent much higher frequencies because the grid is finer, that is all, the definition of high frequency I have changed by changing my grid, but on that grid it is getting rounded and it progressively gets worse, it progressively gets worse, okay, right, so it is very clear this has a, it has a second derivative term that is basically dominating all the other terms, it also has a fourth derivative term and this combination is causing higher frequencies to decay, right, so that you are getting a curve instead of getting a step or a ramp in this case or a representation, you are getting a nice smooth curve, should I try something smaller, you think it will get any better, any worse, yeah, so I do not have that much, I do not have that much hope for this because right off the beginning you can see that it does not seem to help, it does not seem to help, when I made the CFL smaller can you make an observation from last time to this time, is it more dissipated than last time, it looks more dissipated than last time, right, because it had a 1-sigma, right, so as sigma gets smaller and smaller the coefficient of the mu2 term, the coefficient of the second derivative term is definitely getting larger, so I am stressing the code out here, fine, what else do we have, you want to try, there are two possibilities now, maybe what we will do is we will look at the heat equation and this idea of dissipation and dispersion we will look at a little more carefully, okay, so before I do heat equation let me do FTBS itself, I have a little thing here in the bottom where I can add waves, so for instance I can add a sin x, okay, I mean to this I will add, I will make the amplitude smaller, so I can add with an amplitude of 0.5, a wave number of 10, a wave number of 10, so I can add, right, so I am constructing basically what I have done is I have done sin x plus 0.5 sin 10x, okay, so if I run this at CFL1 with FTBS it just goes, in fact because it is 2, I am taking 200 time steps, it is too many time steps, okay, so if I run it with CFL1 FTBS you can actually see that and there does not seem to be, within what you can make out with your eyes there does not seem to be any change in amplitude, nothing is there, it is translating beautifully, translating left to right, what happens if I change the same one to 1.1 CFL, yeah there starts the oscillation, where is the oscillation starting, why is the oscillation starting there, so that is where the high frequency content is, right, I had a sin x but it is not continuing as a sin x, what is coming from the left hand side is a constant function, so you have a function if you think about periodic extension that you would do in sin x, I mean for Fourier series, right, then you have a function that has a, the derivative has a jump, a lot of high frequency terms that will show up there, higher terms that will show up there, it is no longer, once it has shifted, it is no longer just sin x plus sin 10 x, because there is a little straight line segment that is shown up, right, which is going to now include a lot of high frequency terms and you can see that that is there. So it looks like if I were to go to 0.5, let us go to 0.5, I will reset it, we will start from the beginning, what do you say, anything perceptible? It looks like the sin 10 x term, the amplitude is dropping, what about the sin x term? Not so much, okay, not so much, maybe I will add a sin 20 x or a 40 x or something like that, so let us, let us, let us add, I do not know how a 40 x will look, let us add a 40 x and see what it looks like, I have not tried a 40 x, okay, that is, that is, that is needless to say it looks interesting, okay. The graph on the bottom, I am plotting using 800 points, the graph on the top, I am sampling at 200 points, 201 points, okay, so obviously they are going to be sharp for corners, we immediately see the quality of the function is changed and if I take, you can see the 40 x is decaying really fast, right and before you know it, the 40 x is gone, fine and all you are left with is, right, so this sort of moves, this we can possibly explore a little more because this is also simultaneously translating, there are high frequencies supposedly being introduced on the left hand side which is the reason why I have implemented the heat equation because heat equation you have set A equals 0, right, we have the dissipation term but we have set A equals 0, only trouble with heat equation, I mean just one time, let me take one time step instead of two times, this is really dissipative, okay, see two time steps it is going to disappear, so I take one time step that 40 x is still there, two time steps it is almost gone, three time steps it is gone, okay and in this case it is very clear that the high frequency is going faster than the low frequency, it is really decaying faster than the low frequency, is that fine, okay, so if you want to see how many I have done, if I take ten time steps at a time there, right, the 40 x lasts about three or four time steps, it is not there at all, okay, so that is 20 time steps, 30 time steps, 40 time steps, 50 time steps, 60 time steps, that is gone, right and now if you want to patiently wait, if you want to patiently wait, this is going to take a long time, okay, so if you are looking for the steady state solution, let me take a 100 time steps at one shot, this is going to take a very long time, if you are looking for the steady state, okay, so I want you to bear this in mind, this is something that you will note I will recollect, I will recollect this behavior at the end of the semester, right because I need it elsewhere, so yes it is nice that high frequency is decay faster than low frequencies, but if I am looking at the steady state solution, right, if I am looking at the steady state solution and I am looking for the steady state solution and this is the way that the error term is going to go, the low frequency is going to be a headache to get rid of, this is the error term, this is the rate at which my error term is going to decay, the low frequency I have taken, you know, the high frequency is gone and looking for the steady state, the steady state solution in this case is 0, because the heat equation temperatures held 0 at both ends, the steady state equation in this case is 0, so now I am taking in 100s of time steps and it is not going, right, it refuses to go away, is that fine, okay. So I reset, let us try something different, okay, if I take heat equation 0.5 is the stability conditions as 0.5, we are right at right there, so if I make it 0.6, will it make a difference? Let us take one step at a time, well it still seems to decay, not clear, let us try 0.7, reset, so these first few time steps seem to work quite well, let us say wait a minute, what is this, they all seem to work, what is the problem, very difficulty, want me to make it 1, was there a mistake in our stability analysis, so this is another one of those things that you have to be very careful with when you are doing CFD, it looks like something is converging and you may go along thinking it is converging till it starts to diverge, right, say it is converging, it is fine, I am doing great, I have run 5000 time steps, code is converging, it is converged, 3 orders of magnitude, you understand what I am saying, so for engineering purposes that is good enough, this is a classic argument, for engineering purposes that is good enough, no when you are developing your code, make sure it converges, okay, when you are developing your code, when you are making production runs, you have run it for test cases, right, you are making production runs and you have some parameter like A or whatever it is, you run it for A equals 1, A equals 5, you want to try it out for in-betweens, it is worked for A equals 1, A equals 5, then in-between yeah, let it, you know, you can say 3 orders of magnitude, engineering accuracy it is enough, but when you are developing the code, when you are actually developing the code, you want to make sure that it is going to work, if you give it single precision it will work, single precision, if it goes double precision it is going to converge to double precision, you understand, if it does not you want to know why, it is supposed to be stable, okay, and for those of you who may be interested in research, there may be a research problem there, somewhere, so yeah, this seems, it is going to go, you can see it is going fast, okay, it is going fast, pretty pattern but it is going fast, so I say okay, let me quickly try to recover, I go back to 0.5 and that seems to have held it, let me take 10 time steps at a time, so this is like a small struggle going on, right, so high frequency somehow they grew, they took time to grow, they also seemed to take time to go back, okay, but I am taking 10 time steps at a time, let me do make one shot of a 100 here, fine, let us try something different, okay, let me reset this, so that was at 1, I had to push it to 1, would the same thing happen if I did it at 0.6 or was that an illusion, after all 1.1 worked, but remember here there is no advection, it is not being carried out, whatever you have is there, okay, whatever you have is there, so that was it, yeah it seems to decay and just like in the case of, just like in the case of, what do you call it, in the case of CFL of 1, you do not usually call it the CFL for the heat equation, I do not know, yeah there is something, some activity, right, it is really strange, it is really strange and then there is this temptation saying oh maybe just before it, you know blew up I can take that as a solution, there are all sorts of, your mind starts playing tricks on you, right, so yeah, so go back to 0.5 and yes it is, the behavior is about the same, I will take 100 time steps and yes there seems to be some kind of strange propagation come, okay, so here we go, what if I tried, what do you expect will happen if I try something smaller, instead of 0.5 I try 0.1, when we try 0.1 I will take one time step, so as you would expect it is fine, it seems to work, okay, as you would expect it is fine, it seems to work, it is decaying, if I take 100 time steps in one shot, it goes quite fast, okay, I am going to do something little funny now, so what I will do is I will start off with 0.6, so I want to know is this more decaying faster than, is it decaying faster than 0.5, is 0.1 decaying faster than 0.5, I mean for all its heat equation, it should be the same, so what we will do is we will do this 0.6 business, okay, one shot, we are there and then I will change it to 0.1, it goes quite fast, right, okay, so 0.1 is definitely decaying faster than, remember the modified equation, modified equation for heat equation at 4th derivative, 4th derivative is really, really fast, right, especially on those high frequencies, 4th derivative is really, really fast, so it just completely knocked it out, right, whatever you had, it completely knocked it out and now we are struggling with this, okay, the reason why I am harping on this thing is I really want to emphasize this, this is very important for us, now we are struggling, our convergence it looks like will be basically depended on especially if you are looking only at the steady state solution, right, you understand what I mean by looking only for the steady state solution, what I am proposing is, what I am saying is just like we showed that in heat equation, marching in space, marching in time is the same as sweeping Laplace's equation in space, right, it is possible that you are looking for the steady state solution, right, so you have the time derivative and then you start marching in time and you say I want to wait till I get to the steady state solution and there is a small error, I need to get rid of the small error, well this is the small error and the small error is supposed to go to 0, it is not going to 0, okay, fine, before I change this let me do one more because I do not know if you remember this, I will do 0.1666, if you go back and look at the modified equation, right, there was a, that had a 1-6th, right, it had a 1-6th, so let us see what it does, if I take 1-6th, behavior does not seem to be that much different from what I can make out, right, so there will be 6 derivative terms, 8 derivative terms, so there are higher derivative terms, so you just have to see at 1-6th as to whether it is converging faster, not converging faster, okay, that is as far as, let us look at one that I have not done so far which is FTCS 2, I called it, so this is FTCS with the sigma squared by 2 that term added on, second derivative that term added on, so that it explicitly knocks out, it explicitly knocks out the second derivative term in FTCS, which we identified, we pointed to that and saying, oh this coefficient is negative, that is why it is diverging. So first of all, if I do this and I run for a CFL of 1, this should work, it should look like FTBS, it should look like FTBS, what do you say, does it seem to behave like FTBS? Yeah, there is no decay, no nothing, there is no visible decay, nothing of that sort, it seems to behave just like FTBS, is that fine? Okay, what happens if I lower the, what happens if I lower the CFL? Do you want me to raise the CFL and see what happens first? Let us take one time step at a time, I raise the CFL, so yes, you can see that the 40x term first of all was growing but minute enough of it came in from the left hand side is very clear that enough of the right, it is very clear that that high frequency content is also growing. So okay, so it seems to have that behavior from FTCS and FTBS, what if I try CFL.5, look at what is happening carefully, try to think about, give me, I mean think about what I mean, can you make out what is happening, there is something funny happening here, the shape of the peaks and curves, they are changing but they are changing in a unusually slightly different fashion from what you saw earlier, you want me to run it again, I will see, otherwise we will run a smaller CFL, I think then it will become more clear, right, because it is short, there are, look at what happens to those, what happens to those sides, right, look at the, seems to be some kind of rotation kind of a thing going on, is it not? See, I do not know, can you make out? Okay, maybe this does not make sense, so yeah, it is decaying, high frequencies are decaying, it is also translating, very much like FTBS, very much like FTBS, let me try CFL.1 and see whether we are able to get something out of this, CFL.1, so allow me to take 10 times steps at a time, right, CFL.1, 10 times steps, can you see that kind of a, it is not just translating, it is not just decay, there seems to be some, right, some kind of a little rotation like thing going on, is it apparent? Okay, so this is a strange thing, what do you think is happening? This is true dispersion, okay, this is true dispersion, I have knocked out the second derivative term, there is only a third derivative term and higher derivative terms, the third derivative term now has a coefficient that is quite large, okay, but it has a negative sign, so basically what is happening here is the low frequency is travelling faster than the high frequency, the low frequency is travelling faster than the high frequency, so what you would expect and what will actually happen is as we progress with this, you can see that here the low frequency, the high frequency is being left behind, the sine x term and the sort of sine 10 x term are going or chugging along at their own pace, right, it is just that the sine 40 x is sort of being left behind, of course it is also decaying because there is a, it is also decaying, the magnitude is also dropping, right, because the term gets knocked out exactly, only when there is, when there is, you have fourth derivative term here, what do we have, maybe I should count, I will give you do a count, the high frequency there is completely gone and therefore even here it is completely going, I will do it with a count now, so that 40 x, if you want I will add, so by this time, by this point actually the amplitude of the 40 x is almost gone and this is the 10 x, maybe what I will do is I will add a 20 x also, just something in between, right, let me add a 20 x, reset that, we will take instead of 10 time steps at a time, if I take a 100 time steps what is going to happen, it is CFL of 0.1, okay, so that is half the sine wave that is gone, almost half the sine wave that is gone, half the sine x that is gone, okay, the 40 x of course decay is quite fast and the 10 x of course travels reasonably fast, okay, so we have about a quarter of the original sine x term left, okay, so by the time you go through, by the time you go through and the sine x is completely gone, so then the sine 20 x is left behind, am I making sense, is that okay, okay, right, so you can, we can, you could try it out, see this is a case where it is dissipating, we could do the same thing with A of course, we could add a third derivative term explicitly and find out what happens, right, or you could do it with FTCS itself, if you take a small enough CFL, it will turn out that it will grow, if you take small enough CFL it will start to diverge, but before it diverges you can see that low frequencies are actually traveling faster than high frequencies, fine, is that okay, are there any questions, okay, so I think you could try this, let me see what, I tried this with, let me leave it with, let me leave it with 401 grid points, see what happens here and I will take 200 time steps at a time, so CFL is 0.1, right, if CFL were 1, 200 time steps would take me half the way, okay, it will take me half way, but CFL is 0.1, so it is going to 200 time steps is going to take me, now it is going to be too small, you want me to take 400 time steps, this may have other consequences, we will see if it has other consequences, 400 time steps things definitely look a lot smoother, but here I do not know, say I do not know whether in the video it is going to show up, but here you can see that something is traveling backwards, your eye tends to pick up the big motion, which is the sine x, so it look like, because your eye is connected your reference frame becomes sine x, it looks like the rest of it is actually propagating backwards, so if you have to graph it, I do not know whether on the screen it shows, right, it actually seems also something is propagating backwards, because you tend to tie with, your eye tends to follow that sine x, the big feature and the smaller features basically sent to travel back and they are in reference to that traveling backwards, okay, so that I have lost count, so that I have lost count and the whole point of doing this was for me to keep track of how many times I had done it, but anyway it is okay, so yeah the problem with of course larger times, smaller CFLs and larger, what should I say grid size is your time is going to, the time that it takes to do anything is going to increase, okay, so it is very clear that typically right now one lesson that you draw from this is larger CFL, if you are looking for the steady state solution, larger CFL is better than smaller CFL, okay, that is one conclusion that we come to, larger CFL is better to run if you are looking for a steady state solution than a smaller CFL, okay, if you are looking for the transient that is you are looking for a time accurate calculation, right, then your delta T will be based on what is the accuracy that you are looking for within the stability limits, right, but bear in mind that if you just pick up a delta T, you have to ask the question what is this stability, what do I mean by the accuracy that you are looking for, that I am talking from the truncation error point of view, but that is not enough, you can have dispersion and dissipation, I will give you a real life example, okay, give you a real life example, just say your sensors out there basically say oh there goes an earthquake under the ocean somewhere, right, you have used satellites come buoys to figure out what is the height of the wave that is there, okay, over the Bay of Bengal or the Indian Ocean, as a consequence you know that the tsunami may arrive to the coast, right, so if you have, if you have dissipation and dispersion that does not exist in the system, see now I am carefully wording it because the system itself may have dissipation and dispersion, right, remember the bag of potato chips, right, there is always dissipation and dispersion, so if you have dissipation and dispersion that does not exist in the system, then you may mispredict what is the size of the wave, you may mispredict when it is going to arrive, if you mispredict the arrival time, it has consequences, serious consequences, if you mispredict the height of the wave then people may say oh there is this wimpy wave that came in, they warned me, they told me to evacuate, so the third or fourth time they may not evacuate when you tell them to evacuate and making sense, so accuracy is not a matter of saying that oh I have got delta t squared accuracy, you know I choose 10 power – 6, I choose a microsecond or a nanosecond to predict, it is not enough, you have to make sure that there is no dispersion and dissipation because clearly dispersion and dissipation seem to affect the amplitudes that you are getting, they seem to affect, right propagation speeds, propagation speeds which in that example that I gave you will affect arrival times, is that fine, okay, so I think we will take it as a given, now I just that if we wait for this essentially what is going to happen is all the low frequency components are essentially gone and all that is left behind is the high frequency component which will by the way eventually propagate out, in case you are thinking they will also eventually propagate out and you will get this correct steady state solution, right, so if you are looking only for the steady state solution, yes this is the scheme is going to work, it is just that in between you have got oscillations that you are not expecting, is that fine, okay, so there are whole, there are whole host of conclusions that I have come to from these demonstrations, they are all very important, they are going to show up at various times, I am going to use this information at various times in the course as we go along, right, to either introduce techniques or introduce ideas, okay, is that fine, right, are there any questions, okay, so tomorrow what we will do is, tomorrow we will look at the quasi linear wave equation which is more closer to the kind of equations that we are used to, right, I am not going to spend a lot of time on that because I want to get very quickly to the at least one dimensional flow, so that something looks like the Euler equations that you guys are used to solving in your gas dynamics classes at least, right, okay, is that fine, right, thank you.