Lecture 30 : Dissipation/Dispersion Error II

[Music] hello everyone welcome back so we have been looking uh at the for stability analysis in the last couple of classes okay and uh we found that uh for the diffusion problem uh the scheme like ftcs is stable when Alpha is less than half basically the diffusion number is less than half new delta T by delx squ is less than half and for the convection problem what we observed is that the explicit scheme if you use the ftcs it is unconditionally unstable so the central defening scheme uh with an explicit approach is unstable for the linear contion problem okay because amplification factor is always greater than one regardless of what is the Quant number or the CFL number uh that is uh C Delta DX that we are using okay on the other hand when we uh use the implicit scheme what the scheme told us is that uh uh it is unconditionally stable for the linear problem of course when you try solving it uh uh for a nonlinear problem or problem like n equation you will have some more practical consideration uh first of all the implicit is scheme uh is not uh always preferable especially when you are trying to solve a Time accurate uh problem okay and uh so those con will come but what one stability tells you that if you use an implicity scheme it is unconditionally stable whether the problem is diffusion or the conviction okay now we'll look at uh the FBS scheme okay so this is our they uh convection equation d by DT plus c d by DX equal to0 and uh if C is greater than Z okay what I've seen from our experience is that we use the ftvs scheme okay forward in time and backward in space then it is stable for C delta T by DX less than one so this we have seen from our experience right and similarly if C is less than zero okay then we have to use this ftfs okay forward in time and forwarding step I mean this is just to honor the speed of sound so if it is C is greater than zero then we have what you have to do the grid point that you select is I minus one and I to calculate the gradient okay because the wave is going in the right direction when the C is less than zero then you select I and I + 1 and uh to calculate your gradient delx because the wave is going in the left Direction okay so both these cases are called upwinding okay upwind scheme okay because it is one-sided we are not doing Cal defensing and we are using the information uh from the wave to basically find your gradient okay so we'll use the mon stability analysis and see for C greater than Z if our whatever observations we had uh that it is stable for C Delta X less than one is basically valid or not okay then we find out from the one stability indes okay so let's WR the discretized form this is u i n +1 minus UI n by Delta t+ c u i + 1 n minus UI minus one n sorry what I'm using is the first uh FBS right so this will be u i n u IUS n by Delta xal to Z right so use this rearrange it u in + one will become u n minus C delta T by DX okay which we denote by Lambda okay and and then it will be u i n - u i -1 n = to Z okay so just rewrite this as Lambda okay not equal to zero we have already put it on the right hand side okay uh we use our expression u y n will be G to the power n uh e to the power I I and 5 okay u i + 1 n similarly power n power I + 1 i5 and u i n + 1 for example power n the power I I5 so basically uh like uh we have this separation variables so when we trying to solve the differential equation we have basically separated our time and the space okay uh put these things here okay what we'll get is this is g^ n + 1 will Beal to g^ n I'm taking this E power I I5 common everywhere minus Lambda is 1 - e^ - I5 right you can rearrange it this is n is here also n 1 - Lambda 1us cos 5 and then minus minus become plus plus I sin 5 right so that will so now write the amplification Factor g to^ n + 1 by g^ n okay so that you can do 1us Lambda okay and plus Lambda cos 5 and minus Lambda I sin 5 okay just rearrange it so 1us Lambda + Lambda cos 5 okay that's one part minus I Lambda sin 5 okay so this is the value of G okay now it's a again a complex uh uh expression okay so you have this the real part of G and then you have imaginary part of G okay and we want to find out uh for what condition of Lambda uh it will be stable or whether it is stable or not okay that is our task okay so what we'll do what uh we will use the you might have used the complex plots okay uh in your under graduate course okay or in any math course so let's say if you have uh so let's say you have I5 so that is cos 5+ I sin 5 and you can represent it in a complex uh plane okay so you can have this uh real plan okay let's say this is my G so this is my real G and this is imaginary G okay so this has got a radius of equal to 1 okay and five will go all the way from 0 to 2 pi so you can basically this is a circle not made a very good Circle okay just okay so something like this so my radius is equal to one so this is a representation of e^ I 5 okay or is equivalent to cos 5 + I sin 5 okay so if you have let's say e^ 2 II okay so accordingly you can have this uh 2 power5 so it will have radius of two right so basically R cos Theta and R sin Theta these are the two terms okay so now let's look at what term you have so G equal to 1 - Lambda + Lambda cos 5 minus I Lambda sin 5 okay so we use the complex plan so this kind of plot Cal Call ar diagram okay so what we'll do uh we'll take first Lambda equal to 1 okay if you take Lambda equal to 1 this will become G this will become this term will become zero this will be cos 5 - I sin 5 right so when if it is uh uh minus you will uh if you plot from 0 to 2 pi you will have the same thing okay let me just plot it okay so uh the radius is equal to 1 okay so let's say this is my. five this is 1 this is. five this is one this is5 this is one this is. five and this is one okay so uhal to cos 5 plus I sin 5 so we'll have a plot which will connect this origin is at zero right and it will go like this and connect like this okay so this is a circle okay with radius uh is equal to 1 and you can see that its amplitude is one everywhere right so if you take the amplitude of G so this z bar is equal to one okay because this is our amplitude you can take it anywhere this will be equal to 1 okay now let's take Lambda = to 1x 2 okay so then G will become 1X 2 plus 1X 2 cos 5 and then minus I 1X 2 sin 5 okay now here you can see that real axis has shifted by 1X two right so this is uh where our origin will be okay let me use some other color maybe I'll use this color okay and the radius is 1x 2 right so it will have another Circle okay which will go like this will be like this right so this is going uh5 and you can see that uh you can you can take the amplitude anywhere okay the radius is 0.5 but if you take the amplitude anywhere okay so that will be less than one so this is the maximum amplitude is the this which is G is equal to 1 all other places it is less than one right so this is basically stable so one is also stable because that g is just exactly equal to 1 if you take5 then also it will be stable you can take different values of Lambda let's say 75 okay or whatever values you want and you see that it will always be uh inside this circle one Circle unit circle and it will always the magnitude will be always less than equal to one okay so all these are basically G less than one okay now let's go to Lambda equal to greater than one okay let's take Lambda = to 3x2 okay so if you take Lambda = 3x2 then G will become - 1 by2 and plus 3x2 cos half - I 3x2 sin half okay so the origin for this plot also Circle the origin will be - 1 by 2 and 0 right so that is uh basically if I can use another color maybe this one this is my origin I fixed the origin and the radius is basically 3x2 right so now I have to start uh uh make a circle from here and it should go to 3x2 on either side so this is 3x2 okay this is fine -.5 + 1 3x2 here also this is .5 and it has to go uh to min-2 here okay so that will be 1 - 1.5 okay and this is my -2 okay -.5 -1 -2 okay so let's make a circle fixing the origin here and uh use the color that we are using okay so it has to go on either side 1.5 so this may 1 .5 and somewhere here my 1.5 so connect it like this okay okay so this is my circle that will be getting you can easily write a simple matlb code and plot this okay and you can see that uh uh the amplitude of this is always greater than equal to one right it will always be greater than one okay this is basically this is uh uh okay from here we'll connect so this is one and here it is greater than one here greater than one everywhere you will see this is basically greater than one okay so in this case the G is always greater than one right so you see that as you increase the value Lambda greater than one it goes into the Zone where your amplitude factor is always greater than one okay and this is precisely the reason why we find that when we use this value of Lambda equal to C delta T by delx uh greater than one your scheme becomes unstable because your amplification Factor becomes greater than one okay that means any wave uh if you have Lambda greater than one is going to basically uh uh this amplitude will increase in time and finally it will become unstable right so this FBS okay let me write it here ftvs for convection scheme is conditionally stable stable with Lambda = to C Delta d by DX less than equal to 1 okay this is what we have observed earlier also uh while trying to uh basically Solve IT numerically the convection equation that when the Lambda was greater than one it was uh you start seeing vles initially and it will quickly uh basically uh become unstable it will explode okay so this we observed I'll just show you what I have done through this uh plot okay so this is what I was trying to show you okay so this is Lambda equal to.5 okay and you can take this amplitude anywhere okay this will be less than one this is Lambda equal to 1 okay uh and it will be one everywhere and then we have limal to 1.5 okay if you can use this color okay this is uh greater than uh one okay and because of that it becomes unstable so you can uh when you go back please uh uh work it out and see plot for different lambdas and find out when becomes stable and it becomes unstable okay so similarly you can uh try this uh uh uh this one stability analysis for many different problems okay and I'll not go through all of these but I would uh uh basically uh request you or basically ask you to try this as a practice assignment okay so you can take different schemes that we have discussed in the class so you have this uh uh ctcs scheme Li frog scheme okay so which is uh uh like the uh Central defensing in space as well as Central defensing in time so both has got Central defensing uh space in time and see whether this scheme is stable or not okay using the volum stability analysis similarly look for The Lex fedc scheme which is basically ftcs but now we have the UI term is taken as the half of the I +1 and IUS one okay and then you have the LAX vand of scheme which has additional diffusion on the central defening scheme so use your for okay U equal to uh u n is equal to ^ n e to the^ I i5 and C what stability characteristic you find for these equations are there any conditions where it become stable and try it for conviction problem okay this is for conviction problem similarly you can try so this are already tried implicits we have already done that okay uh try for the crank also we have discuss the SCH also and find out the stability conditions for all these schemes okay so this is the homework for you and we'll uh uh continue discussing it uh uh as in when necessary but uh this is something we should be very very uh basically you should be convenient with the stability conditions okay uh it should not uh uh be appearing that uh uh you are using uh the central defening scheme without knowledge uh to a contion problem okay so if it gets in Grand uh basically shd knowledge that uh if uh the if the Lambda or Alpha what are the conditions uh when it can be stable when it can not be stable what the schemes which are stable or not stable it will become very useful for you okay uh while choosing your scheme for a particular problem okay so what I'll do in the next class so we have done the stability analysis now we have to go for something called error analysis okay so what is error analysis so right now we have only talked about whether the scheme is stable or not and from the L from you we know that if the scheme is consistent and stable the convergence is guaranteed but when if it is conversed okay depending on what scheme you're using you will have uh basically uh certain error okay so the errors can be of two types okay one is the dissipation error okay and other is dis disperson error okay it also sometimes called diffusion error although there are some differences uh between the dissipation and uh diffusion okay but they are both related to similar terms okay so we will be alternately be using uh this disp error the disp error the disp error is also called the phase error okay and this can also be called an amplitude error okay so uh we will look at uh these two errors uh because that will tell us the information about uh which method is more accurate okay we have already discussed about the the order of accuracy earlier okay so the dissipation error will basically we can figure out from our order of accuracy so this is basically order okay and this disperson error which is also very very important if you are particularly interested in uh basically High fidil flows okay so uh we'll discuss what is this person error uh so this is related to the resolution of the scheme okay so dis eror is related to the order of the scheme and dis error is the related to the resolution of the scheme so what does it mean so dissipation error so let's say you are trying to solve a problem which is like this okay and if the scheme has got dissipation error so what will happen with time okay as you solve it so let's say you're trying to uh use your convection equation okay plus c equal to0 okay so it should basically ideally just propagate like this okay and then go like this okay with the speed which is C okay but what happens actually is that it propagates but also you will see that the magnitude comes like this even it can with time it can even go lower okay so for example it is going like this okay we have already seen that right and further if you go down then it will probably go like this okay so you can see that the amplitude is started going down so what started with this uh big amplitude now it has become like this okay so if there is a uh change in the amplitude okay uh particularly DK of amplitude that is called the dissipation error okay so this is our dissipation error okay now what is disperson error okay let's look at the same wave and we are trying to solve it uh uh using uh let's say the central differencing scheme okay so this let's say we're solving ftps scheme okay now let's say we trying to solve this using ftcs scheme okay Central differing which we know that it is unstable right so so the exact solution will be something like this okay okay but you'll see that if you're using the central differing scheme okay so you will start seeing Wiles like this okay the amplitude may not go down much but we will have Wiles let's say here and here you may have Wiles here also here also okay similarly uh it may you may see start seeing wiggles at the discontinous region okay and ultimately what will happen if you continue running it then it will blow up okay the scheme will blow okay so what you are seeing here in this compared to the disp error so in the dissipation error it is propagating uh but it is basically the magnitude is going down okay or maybe it is smearing out okay but in this case uh the amplitude or magnitude is not going down but you start seeing some oscillations here okay like here okay so what is exactly happening is that let's say we we discuss that this Square wave in this particular case consists of many waves okay many let's say s or cosine waves okay we have already discussed the for representation okay now when you have the disperson error so uh so different waves which means different frequencies okay start moving at different speeds okay and that basically depends on the it starts depending on the wave number okay so if you have a square wave I if you're just solving this problem d by DT plus c d by DX you expected that all the waves let's say this Square wave or any wave that any input wave that you have consist of many different waves so all those waves will travel at the speed of C right that is our expectation okay but what if some waves is start moving at a speed greater than C and some W start moving at a speed less than C okay and what if they depend on the wave number so the representation will be uh basically let's say sin KX okay k1x k2x k3x that's so many different waves right so what if the speed starts depending on this K okay so that is basically completely numerical okay it is because of your choice choice of scheme your different waves are traveling at different speeds some are basically becoming greater than your expected speed and some are lower than the expected speed okay so when the wapes are not traveling at the expected speed that is called dispersion okay so this phenomenon is called dispersion okay and the error due to that is called the disperson error okay so this is a very very dangerous error okay uh because if you do not control the disperson error what will happen is that your scheme will eventually blow up okay at the same time so if you take use any disc scheme it will have error okay because you trating the T Series okay so it can have the disp error or it can have the dispers error right now if you want a scheme uh which is uh very good for solving a problem you want to minimize both these errors right so if you have the displac an error as the leading order term okay what will happen is that your you will not get unstable solution uh you'll get some solution but it will be your amplitude will be basically modified on the other hand if you use a uh equation which has got a disperson error in the leading order term your wave number will be uh basically modified okay so now our art uh in cfd is to find out a scheme which has got minimum of these errors okay and at the same time you should uh basically ensure that your dissipation error is slightly or maybe in in such a way that it basically uh mitigates your dis disperson error okay so basically it smears out and it gives you a stable solution if the disperson error is in the leading order term okay then and it is quite possible that your solution will go up okay so we'll have a more systematic discussion on both these errors in the next few classes and we'll understand like we have used the one stability analysis and understood the stability we'll do error analysis and find out the errors in different schemes okay uh we have discussed about the modified PD and that will come into picture when we start discussing uh these errors okay so with that we'll stop today's class and'll meet in the next class class thank you very much [Music] [Music] [Music]