Okay. So let's get on. So what, what we're aiming to do in this segment is get some sense of the stability of the equations that we need to look at. Now if we want to look at stability we have to first understand we must first understand the stability of the time exact case, all right? Because that is the, the sort of behavior, the sort of response we are aspiring towards for our system, right? For our algorithmic system. Okay? So, in terms of stability, let's first understand the time exact case. Right, now, we've derived the single degree of freedom, modal equations, for a partic, for an arbitrary mode L. Okay? Now, everything we do holds for every mode, right? The, the, the, our, our analysis holds for any mode, because we're really working for an arbitrary mode. With that in mind, I can afford, I believe, to drop the explicit, use of the modal index, L. Okay? All right, so I'm going to drop that, right? So, from now on, the time for, for the time exact case, since we're working a single degree of freedom, for the time exact case and for the time discrete case, when we are working with single degree of freedom modal equations, I'm just going to write them as d dot scalar plus lambda d equals 0. Okay? But one more thing, I've got rid of one index L, but I want to bring another one. The one I want to bring back is the, I'm going to use h, right? And it's not an index, it's a superscript. Why am I bringing h back here? Remember h is our old friend the element, size. Right? It denotes the element size. The fact of spacial discretization. Why am I bringing it back? Right. It's because our n and k matrices depend upon our discretization, our spatial discretization. Right? So the ei, eigenvalues we're working with here are truly the spatially discretized eigen, the, the eigenvalues corresponding to the spatially discretized system. Okay? So lemme, lemme just state that here. Lambda h is the eigenvalue of a mode, or corresponding to a mode, that was, that is obtained after spatial discretization. Okay, so the factor spacial discretization, which is which shows itself up, which, which shows up in the finite element size, is indeed reflected in lambda h. Okay? Alternately, because these are partial differential equations, one could do a fully continuous analysis of them, and then one would have a eigenvalue corresponding to modes, but those would not be discretized modes, okay? Those modes or those eigen those eigenmodes would be actually eigenfunctions, not eigenvectors. Okay? So there would be a, so there's a difference. We're really working with the eigenvalues of the, of the spatially discretized system here, and that will show up, it's, it's for that reason that I'm bringing back our memories of h here. Okay? All right, so the time exact case is this, plus of course boundary condition. Sorry, the initial condition, right? So this in fact, d(0) equals d not, right? On the previous slide we've used the modal index L, but we've just decided to drop it here. Okay? Without, without risk of confusion, because everything we do holds for every mode. All right, so what is the stability of the system? How do we, how do we know what the stability of the system is? This equation is one of the simpler ODEs you're likely to encounter. Okay? We can directly write down the exact solution. All right? So the exact solution is d as a function of t equals d sub 0 exponent of minus lambda h times t. Okay? It's easy enough to check that that is indeed the exact solution. If you plug it into your, equations, you will find, and into our ODE, you will find that it satisfies the ODE, and it also, respects the initial condition. Okay? Just set T equal to 0 on the right hand side, since exponent or minus 0 is 1. You get back d at 0 equals d not. Okay? So this is the exact solution. All right, now, what about lambda h? What do we know about lambda H? It's an eigenvalue of the system, right? What is lambda h? Lambda h turns out to be greater than or equal to 0. Okay? All right? Why is lambda h greater than or equal to 0? We're not going to prove it, but do you know what properties give us this? It is the fact that M is positive definite. Right? And K is usually positive definite, but in the most general case, if we, if we want to also allow for insulation along certain directions, if you're talking of heat conduction or the possibility that there is no transport along certain directions if you're talking of mass diffusion, then k is a positive semi definite Okay, earlier on we'd used the fact that K can pretty much be taken to be positive definite, unless you really want to have insulation. We'd use that fact to make the observation that arriving that, that we are going to get eigenvectors that, that are linearly independent. Okay? All right. So, we have this, we have this sort of situation. All right, if that is the case since lambda h is greater than or equal to 0, what can we say about d? Say tn plus 1, right? And we know what this means. It just means that we're evaluating the solution at time n plus 1. What can we say about d at tn plus 1 relative to d at tn? Right? And, and here I'm using the fact that because of the nature of our time discretization and our choice of the progression of time instance, tn plus 1 is greater than tn. Right? So, let me just recall that also. We are, of course, using here the fact that we are progressing in time, so tn plus 1 is greater than or equal to tn, right? It's usually, it's, it's greater than tn. We nev, we never use it equal to tn, because that would mean we have a 0 times. Okay? All right, so given this, what should I use in this blank here? I've left a big blank spot between d and tn plus one and d and tn. What relational operator do I use? Right. Because the exponent of a negative argument is less than one, right? What we see is that this is a decaying function. Right, it's monotonically decreasing. Okay? Or another way at looking at it is, that d at t n plus one divided by d at tn is lesser than or equal to one, provided of course d at tn is not 0. Okay, but, nevertheless it's monotonically decreasing. All right, so we have monotonically decreasing, sorry, time dependent coefficient for our mode. Okay? All right. And this essential says that the nature of our heat conduction equation or our, or nature of our, mass diffusion equation, the kind we are looking at here, is for the solution to tend to be K. Okay? There is no tendency for the solution to tend to increase, provided we have set the forcing equal to 0. All right? And it is on in order to expose this characteristic of the equations we are working with that we are considering the homogeneous case. Because clearly, you could be supplying heat to increase the to, to, to raise the temperature, or you could be, you could have a local supply of mass, or, or, or, again, influx of heat or mass in order to push up the temperature or the mass concentration at, at any point. And therefore, these modes could be increasing in a problem that is in homogeneous. Okay? But we wanted to get to this fundamental characteristic of the equation, so we just assume the homo, we, we are considering the homogeneous case. All right, so why are we doing this? The reason we are doing this is because we want to understand what is the exact behavior that our algorithmic equations should aim to represent? Okay? All right. So, with this in hand, let's ask, ask ourselves what, what the same sort of analysis tells us for our time discrete equation. And this is really quite easy, because the way we've written out the time discrete equation, it is, it is in algebraic form. It just gives us dn plus 1 multiplied by co, by some factor minus dn multiplied by some other factor, okay? So the time discrete equation, if you go back and look at it in your notes, and I have it right up here in my slides, so let me just flash it up again, it is in the middle of this slide, right? Marked out [INAUDIBLE] as time discrete equation, time discrete case. Okay? So I'm going to rewrite it using the same notation that we are now following, which is to drop the modal index L, and instead for lambda, bring back the superscript h. Okay? So the time discrete case we'll use in this notation is the following. It is d times 1 plus alpha delta t lambda h equals, I'm just moving it to, to the right hand side, equals dn 1 minus 1 minus alpha delta t, lambda h. Okay? Sorry, and here I have dn plus 1. Okay? All right, now, let's try to find that same ratio that we'd found in the case of the time exact problem. All right? So, the equivalent ratio to form here is dn plus 1 and d divided by dn. Okay? And this as we see is equal to 1 minus 1 minus alpha delta t lambda h divided by 1 plus alpha delta t lambda h. Okay? Now, in the context of our time discrete algorithmic problem, we tend to call this a ratio, we tend to denote it as A. Okay? And A, very obviously is picked because we really want, because this is really an amplification factor. Right? Inasmuch as it is obtained as a ratio of dn plus 1 at, over dn, it essentially tells us how is our time discrete solution getting magnified from one time step to the other. All right? Or getting amplified from one time step to the other. Okay? And you note that A depends upon, if you want to think of it in that way, it depends upon alpha, it depends upon lambda t, and it depends upon lambda h. Right? Okay? So, what that says is that, yes, our integration algorithm matters, right, whether we're choosing one type of the oiler fam, one member of the oiler family or the other. Our time step matters for stability, that's not, that should not be a big surprise. But interestingly as well, the spatial discretization we've used does matter. Okay? And this is why I took pains to point out that the eigenvalue we're working with for any mode, is really a discretized eigenvalue in the sense that it, it reflects the spatial discretization. All right? And that too does affect our amplification factor. All right? When we, well let's just do one more thing. What do we mean now by a stable problem? Right? We wanted to reflect what we saw for the time exact case. Okay? So, for stability the requirement we want to have, right, is the following; we want to say that the magnitude of A should be lesser than or equal to 1. Okay? All right. Now, you may wonder why we are going with the magnitude. All right, what would happen, and why not just say, a has to be less than one? Well, what you're going to see, is that, this is a, after all, we're doing approximations here, right? We're constructing the, the reason we're the getting districtizations is we want to approximate the time dependent behavior. Well, one, result of that, approximation and of that districtization is that it possible for our solution to sometimes go negative here. Okay? So that is something that we will have to deal with. Right? And that's why we recognize that A could be, could, could be negative, dn plus 1 over dn could be a negative ratio. Solution can change signs, we're just recognizing that the algorithm may do that and therefore we are restricting ourselves further by saying that the magnitude of A has to be lesser than or equal to 1. Okay, we are not guaranteed, positive solutions always for our time discrete problem. Okay, so when we come back, we're going to end the segment here, when we return, we are going to, apply this, this condition to our, time discrete, problem and see what it tells us.
