so we ended up with our clean version of the discontinuous lurking formulation and what we had to introduce a couple of new domains Omega tilde gamma being are broken element or a broken domain volume and our mesh skeleton and we had introduced these two new operators the average and the jump and and this jump was a bit of an unintuitive definition that actually ended up cleaning a whole bunch of of the mess with the the pluses and minuses on different sides of element interfaces and then we said well we have to choose now a numerical flux and we want to have certain properties of this numerical flux and these two properties are called the consistency and uh flux is consistent and a flux is conservative so how to choose how to choose the assets finally and sadly I'm still going to have to introduce five pluses and five minuses U so we want them to be consistent and we want them to be conservative what do these two terms mean conservative well they have pretty straightforward definitions so I'll write those down and then we're going to talk about what that means the consistency definition is that um F hat of Phi comma PSI is going to be equal to F of Phi and the conservative conservativity of a flux is of a numerical flux is the property that the jump of f is equal to zero and we often call this a single valued flux okay so why do we want to have these two properties so let's first think about the consistency and maybe you find this a very obvious property um because what this is saying is that if our solution is actually single valued right so if we have a field Phi that does not have a plus or a minus sign but if it just has a single value on an edge then our numerical flux F hat should equal our true flux F evaluated with that diamond yeah so that that is rather straightforwardly a property that's going to ensure that we have um we had the same type of property for our residual based uh stabilization methods and that we want to make sure that the true solution actually satisfies uh the finite element formulation and that was called variational consistency and this also has to do with with the same line of reasoning you know so a consistent flux is going to lead to a variationally consistent discontinuous method which again means that if we actually were to obtain uh the the true solution Phi well then it would satisfy our finite element formulation and typically we don't obtain the true Solutions but we certainly don't want to develop methods uh that would not be satisfied by the true solution right so that's a rather intuitive notion and actually uh more practically or more specifically um consistent methods are the ones that also converge and that also makes sense if you don't have a consistent method if your method is not satisfied by the true solution then naturally if you mesh refine you won't converge to the true solution it also precisely that is guaranteed by making sure that the numerical flux is the one with the hat equals the true flux if we have a single valued uh solution field okay then the conservativity now this one might be slightly less obvious this is saying that the numerical flux evaluated on either side of our boundary is single valued um that's going to guarantee that we have certain conservation properties in our solution that if we solve for this continuous method then that solution is going to satisfy certain uh conservation properties and we can actually show this now with our cleaned up formulation so assume that I have a flux that is conservative well that actually simplifies our formulation yet again because that means that this guy is equal to zero and we're only left with this so at that point I'm going to call um the average of the flux F hat is simply going to be equal to well F hat now I don't have to make the distinguish anymore okay I feel like I'm skipping something that's why I'm going a little slow here yeah okay maybe I I wanna point this out of course my fraction valuation I've had depends on while the values on both sides of the element but I could also for some strange reason have a different flux definition on either side of the element right so you would also have an f plus or minus and we'll actually see that some people have proposed this so we have a plus or a minus and that's why we can define a jump right and that's why the jump is not necessarily zero that's how I felt like I was getting a little bit uh felt like was a bit sketchy of why uh the the numerical flux might be zero at all so that's why I'm trying to that's what I'm trying to explain right now um so if we have a different flux definition on the the the one-sided the element and the other side of the element and then you would have a non-zero jump so a conservative flux is precisely flux for which we do have a zero jump which means that well we don't have a distinguished between the plus and the minus side and that's why I'm dropping that plus minus right here but we'll actually actually uh later on see examples of fluxes that are not conservative and do have different definitions on two sides of a boundary um now in principle that that's actually uh going to introduce these annoyances of uh having to take very good care about which element is the plus side and which element is the minus sign but we can actually uh still put those also into a framework that looks like a formulation right here how about I'm getting ahead of myself we'll see that let me see that foreign so what I wanted to talk about is why we're calling this a conservative flux so why is this property so important for making our solution be conservative so let me um rewrite this equation and then we're going to manipulate that a little bit so now we have again integral of Phi TZ until the minus f uh dotted PSI multiple the gradient of V plus and this again also integration over a broken domain and then we have a single integration over our interface skeleton as well our single valued and numerical flux multiplied by the jump of G or dotted by the jump okay so I know that once I have run my discontinuous lurking simulation and I have obtained a solution then I know that my solution will satisfy this equation for all possible choices of test functions for any possible choice of test function so what I can do is I can take this equation I'm going to say well apparently my solution Phi is going to be set aside for a test function that looks like this suppose that we have a mesh and I'm going to choose my test function V to be equal to 1 inside one element and zero in all other elements well again since my solution satisfies this equation for all possible choices V V in our space that means that it is also satisfied for this choice where I have a one inside one element and zeros everywhere else and I'm only allowed to make this well I'll get back to that later okay so suppose that we choose this function V then what does this statement end up saying well we have now an integral only on our elements right only on this single element so we have an integral Phi t time derivative and well it's going to be integral multiplied by one so I'm simply have the integral of that that quantity so how about the second term well I have dotted where the gradient of V inside of the element well the gradient of Z is going to be zero everywhere so my second term cancels how about my third term well this third term it's gonna be zero on all uh parts of our interface skeleton right because everywhere my jump is zero except for on the boundaries of this element so I have an integral over the boundary of my elements where the V on the one side is going to be equal to zero and V on the other side is going to be equal to it's not going to be equal to zero so let's at this point I'll have to introduce again the sign so I'll call this plus this minus and any one of these so now we're left with F at um Dot N plus V plus and since now I'm only interested in a single set of elements and I'll actually drop these pluses and I'll simply say this is going to be true for every possible choice of of Adam because I can always set my as my test function equal to one one element and zero everywhere else and this is precisely what our original um conservation law was about right so this is saying and actually this is equal to zero as the final statement this is saying that the time derivative of our total quantity of Phi h is going to be equal to minus partial k F hat I'm sorry V is equal to one so this terms and that's precisely the conservation law that we started off with and so and we can only do this because of this um this single valued nature of the flux function because if it was in single value we would have this other term right here we would have some additional term that's going to hinge on the flux um on both sides of the element yeah so you would get a rather funny relation um dictating the uh the flow of Phi the the way that Phi is going to distribute right but right now we have that and amount of Phi the time derivative of PSI the amount of in which Phi is going to change is going to be only equal to the the flux through the boundary and since we also know that we have a consistent flux or that's that's going to match up very nicely with how we want our partial differential equation to act right we're dealing with hyperbolic conservation law and this is precisely a conservation law set on one element and that's satisfied in the case of a conservative flux and that's why we're calling this a conservative flux because now it satisfies the conservation law on each element um and one other thing that I think is very important to realize here is that we can only make this argument because of the discontinuable argument now we would not be able to make this argument in a contentious method because you cannot choose a test function that looks like this oh you cannot choose a continuous test function that is one inside a single element and zero everywhere else right this is a discontinuous test function and what this continuous functions are not in a continuous lurking method and so at most in a continuous lurking method you could choose a test function as one at a node and then zero everywhere else but that means that it has a gradient here a gradient here gradient here grain here will be non-zero and each one of these elements it would get a very messy messy quote analy quality would still be satisfied but it wouldn't be a nice clean cut conservation equality I like the one that we have right here okay good um and we made this argument before but I'll reiterate this uh the the reason why we're calling this conservation equality uh is not necessarily because of this equation itself but it's because we can write the same equation not only on this element but also on this element and then we would see that whatever flows out of the one element is going to flow into the other element through this boundary and again that that is precisely what would not be the case if we would also have this term they will get some message message stuff in here that's going to not satisfy that whatever flows out of one element flows into the other okay and I think that's all I wanted to do for this video um so we haven't talked about what to actually choose for these flux functions and we'll do that in the next video but now we have an idea of what we want our flux functions to look like we want them to be consistent and because that's going to ensure that our we have convergence that behaves the way it should very simply put and it it we want it to be conservative because that reflects the physics of our um of yeah we want that reflects that make sure that the discontinuous glucan method uh reflects the physics of our relevant partial differential equation okay then uh I think that's it for this video and in the next video I'll actually prepare a bit of a table with different examples of uh of numerical fluxes and how also connect back to this and and see if these are conservative and consistent or not okay okay see you in the next video thank you for your attention