Okay, so what we will do is; in the last class
we saw a demo of the one dimensional Euler
equations, right and we were looking at various
behaviour, there are certain things about
the demo that I just want to recollect here,
one we already know that there are 3 different
propagation speeds right, so in the demo I
took a constant time step everywhere, it did
not look like right, I took delta t/ delta
x, I kept it a constant which basically means
that I was taking delta t to be a constant.
In the demo, I took delta t equals constant,
so in the sense when I say delta t equals
constant, I mean delta t equals constant,
it should say delta t is in a sense homogenous
that is delta t is a constant, this may not
make sense to you, delta t is a constant across
grids, right just all over grids. The other
thing that we saw was that these waves were
propagating, we saw that there were compression
waves not necessarily corresponding to u +
a or propagating at u + a.
The compression waves possibly corresponding
to u + a propagating at higher speeds one
way and rarefaction waves coming in the opposite
direction, right which were taking more time
because they are propagating something like
u - a so, because the characteristics are
u, u + a and u – a, those the characteristics
are u, u + a, u – a, it is very clear that
as we get close to convergence, one of these
may become very small, right.
So, just to recollect what we did in the demo,
we had a simple pipe and we had a pressure
here P0 and initially because the flow and
the pipe which is of some length L and you
have K ambient on this side, I will indicate
T0 here because we have the boundary condition
but I am really interested in the pressure
coming to equilibrium in this case; in the
demo, so if you recollect the demo you had
a compression wave that propagated left to
right.
Of course, if you have a discontinuity you
know that you have to take the left conditions
and right conditions and right, to figure
out the speed, there is a specific way by
which we can do it but we will just say that
it corresponds to u + a, right and those contact
surface that was essentially traveling at
the speed of u and when this T0 was communicated
from this end to this end because it is a
compression wave bordering on a shock, it
was traveling quite fast.
It propagated through this and when the P
ambient condition which were in enforcing
here caused an expansion fan to travel upstream,
you understand what I am saying, so there
is an expansion fan traveling upstream which
is propagating at u - a and as u gets larger
and larger u - a is going to get smaller and
smaller, so it is going to take more and more
time to propagate upstream.
Now, it is clear that when you get convergence,
if you are looking for a steady state solution
these static pressure here and the static
pressure there have to come to terms in a
sense, they have to come from some kind of
an equilibrium in this; the reason why I took
a constant area duct is that they have to
be the same, the P0; P in fact will be a constant
throughout that is the solution.
So, you are going to see this reflection back
and forth constantly, right and if the back
part is going to take more and more time as
time progresses, convergence will take more
time, so one of the observations that I have
is; I have taken delta T constant and convergence
can be slow; convergence can be slow, so if
u - a is small, this happens when near transonic
flows, right, so near transonic flows, u - a
is small.
The other is very low subsonic flows; low
subsonic flow, so that transonic flows; low
subsonic flows, u is approximately 0, u is
very small, so any time these propagation
speeds are small, we are going to have difficulty,
right and I ended the conversation in the
last class by just mentioning the idea of
stiff differential equations, right, I mean
you use the word stiff, which is a structures
kind of a term.
So, you apply, you imagine that if you have
a stiff beam, you apply an extremely large
force and you get a small deflection, it is
very stiff right, so there is equilibrium
involving very large entities and very small
entities that is the idea, right so it is
very stiff and those problems like that can
sometimes be very difficult or very often
can be very difficult to solve and in this
case, I am pointing out only a very simple
scenario.
Sometimes, what happens is; as in the case
of u approximately 0, you could even get spurious
answers, you can even converge to answers
that are not the right answer, right based
on what is the algorithm that you choose okay.
So, there is an issue of slow convergence;
is an issue of slow convergence, there are
2 things that I want to take from this demo;
take forward from this demo, one is how to
speed up convergence, right.
The last time we talked about speeding up
convergence was when we are talking about
SOR okay that was the last time we talked
about speeding up converging, the other we
will look at now; we look at some mechanism,
we have actually looked at what do you call
it okay, this fast and slow we will talk about
it, we actually looked at other schemes where
we tried to speed up convergence.
The other thing that I want to look at is
getting actually, a time accurate solution,
so far we have been talking about steady state
solutions, is it possible that I can actually
calculate the transient okay, right, given
that we kept this delta t constant, so we
were sort of evolving an idea Runge kutta
method, which is a higher order scheme and
time right, so we will see; we will say; we
will try to say something about getting time
accurate schemes.
Something about what I mean by slow convergence,
so it is not enough see, you can do the math,
you can look at algorithms, you can say oh,
this is an order and squared algorithm meaning
that if you have something like a matrix which
is of size and right, that the time that it
is going to take to compute is going to be
n squared, it is possibility, right, you have
all of these order of magnitude calculations.
You have; you can be given any number of guarantees
by the algorithm, by the programming language,
by the machine that you have people can make
all sorts of claims but finally what counts
to you is what I would call the wall clock
time. I sit down, it is 9 o'clock, I start
running my program when do I get my answer
back, you understand. We can stand here waving
hands, talking about floating point; number
of floating point operations, gigaflops, mega
flops all of this kind.
So, it does not help, those numbers do not
help, what really matters is there is a clock,
you have a question, you start your program
running, when will you get your answer back,
okay and if it is going to take a long time,
if it is going to take a day that may be considered
slow, if it is going to be taken a minute
that would be considered very fast, right,
it depends on what your application, depends
on the size of your problem.
For a given problem right, obviously something
that takes few days is slower than something
that takes a day; wall clock time, am I making
sense okay, so in that context when we did
approximate factorization, we are looking
at an acceleration scheme. If you are looking
at something that takes less effort and you
get the answer back quickly, we were actually
looking at when we did approximate factorization
for implicit schemes.
We are trying to look at ways by which we
could make our program run faster from the
point of view of wall clock time, when I say
faster, I only mean from the point of view
of wall clock time, so you have looked at
2; SOR and some approximate factorization
schemes and so on. So, we will look at another;
we will look at another class of schemes,
we will look at one now and the little later
I will introduce other ideas to talk about
how to accelerate convergence and so on okay.
So, we will try to do this first, I will set
this up and then I will just say something
about unsteady flows, fine. Since, I have
talked about approximate factorization, we
recollect that basically, we had some term
if you recollect if we had some term going
to 0, we said what is the big deal, what happens
to the coefficient, we are going to use the
same idea here. What did we say we have very
disparate wave numbers, we have very disparate
wave speeds, I am sorry.
We have very disparate wave speeds, so if
you have dou Q dot t + dou E dou x equals
0, one dimensional flow, you have propagation
speeds that correspond to u, u + a, u – a,
where a is the acoustic speed okay, if we
are looking only for the steady state right,
only for the steady state solution seeking
steady state solution, if you are seeking
only steady state solution, what is going
to happen? Dou Q dou t will go to 0, okay.
When I say I am seeking steady state solution
that is I am proposing to use a time matching
scheme; I am proposing to use a time matching
scheme to match these equations in time to
steady state that is the objective and if
I were to match these equations to steady
state, dou Q dou t would go to 0, okay that
is the opportunity for us. The minute we see
dou Q dou t will go to 0 that gives us the
opportunity.
So, I say using literature; the notation used
in literature out there, why not multiply
by a; pre multiply that term by a matrix gamma,
okay pre multiply that I say pre multiply
this term right by matrix gamma because I
know that this quantity is going to go to
0, right so how does it matter, I just pre
multiplied by gamma, fine. So, as a consequence,
this equation having done this, so I propose
now to solve this instead the original Euler
equations.
If I pre multiply this by gamma inverse, what
do I get? I get dou Q dou t + gamma inverse
dou E dou x equal 0, this tells me that you
have dou Q dou t + gamma inverse A dou Q dou
x equals 0, fine and what we want to do is;
we want to choose the gamma, so that these
eigenvalues are nice, we want to choose the
gamma, so that the eigenvalues of that matrix
are nice, is that fine okay.
I want to choose the gamma, so that the eigenvalues
of this matrix are nice and keep life easy
but the eigenvectors are the same. Eigen vectors
are same as that of A, fine, is that okay.
So, what I am saying is; so what would be
nice Eigen values, they are all says the same
order 1, okay so the nice eigenvalues would
be; the eigenvalues that we have right now
are u, 0, 0, 0, u + a, 0, this is the matrix.
This is the diagonal matrix that we get, what
I propose is that you; we build something
that is I will put a 1 there, which is they
are all the same magnitude, am I making sense,
I propose that we take something like this,
fine and having the same eigenvector.
So, what I am saying is that X inverse AX
equals lambda, what we normally write is AX
equals X lambda that is what you normally
write or what is the other possibility that
I suggested now, X inverse gamma inverse AX
equals lambda 1, okay or gamma inverse AX
equals X lambda 1, is that fine, okay. What
do I get now? Maybe I can substitute from
the AX here, so I get gamma inverse AX X lambda
equals X lambda 1.
Yeah, maybe I can multiply through by lambda
1 inverse; gamma inverse X lambda lambda 1
inverse, what is this product; equals X. So,
in our case right now, this looks like mod
lambda, this looks like that you can just
check this out which is mod u, 0, 0, 0, mod
u + a, 0 , 0, 0, mod u + a, so 
is it possible for us to find gamma from here,
how do we find gamma from here? Excuse me,
so this may not be the greatest way to find
gamma.
There are different ways by which you can
do it, I just wanted to give you a clue, so
you can work this through, it is actually
possible for us to solve for gamma now and
once you have solved for gamma, you come back
you can solve that equation, am I making sense
in the propagation speeds will be almost equal
okay, so there is something that you can try
out, so it is very easy that you; this is
as I said this is a very naive method.
This has been tried now for almost 25 years,
people have been fiddling around trying to
figure out various values of gamma that will
give you, right extremely good convergence,
this process by which in any stiff equation,
whether it is a linear system as in algebraic
equations, right or differential equations,
any stiff equations where you have very disparate
eigenvalues and you do something.
Typically, you pre multiply by some matrix
or pre multiply by some operator right, this
process where you do that, so that the eigenvalues
become nice, it is called preconditioning,
it is a big idea, so numerical analysis we
use it quite a bit, it is called preconditioning,
right.
It comes from, in matrix algebra I do not
know if you are aware of this, the ratio of
the largest eigenvalue to the smallest eigenvalue
is called the condition number, just to give
you an idea as to where it comes from, in
matrix algebra condition number is; I will
just say rho A/ rho A inverse given that A
is invertible, okay, so it gives you the ratio
of rho A times rho A inverse, anyway, rho
A times rho A inverse; A inverse will have
rho A times rho A inverse, is that right.
So, I am going to make it up real time I have
to be careful okay, fine, given that A is
invertible okay, largest Eigen value/ smallest
Eigen value okay, is that fine and if the
Eigen values are very disparate, this ratio
will be very large, it is called the condition
number, fine and we will see, we will keep
on coming back to this, this is a big headache,
we will keep on coming back to this, a lot
of the acceleration schemes that the other
set that I am going to look at will deal with
the same idea, okay.
We will deal with the same ideas, in this
case because we are multiplying the unsteady
term it is called preconditioning the unsteady
term okay, so in this case we are preconditioning
the unsteady okay, is that fine, right now
what I will do is; as I said there is another
class of acceleration schemes that we are
going to talk about, I will get to that later,
since I am right now talking about fiddling
around with the unsteady term.
And simultaneously, I said look I am going
to tell you how to do time accurate calculations,
we went to the effort of taking a constant
delta t everywhere, I want to say something
about the consequences of this okay, is that
right okay. So, how do we calculate; how do
we get time accurate computations, what are
the issues involved, I am not going to spend
a lot of time like as I mentioned, there are
lot of these topics that I am just going to
give you enough that you have a flavour for
what the topic is about, right.
I am not going to spend a lot, there can be
a whole course on unsteady aerodynamics and
then consequently a computation of unsteady
flows okay, so but what we do here is; if
you have this equation, we have already seen
there are 2 issues that are involved, so it
is not enough that I use Runge kutta method
right, to integrate in time, it is not enough
that I just use Runge kutta method to integrate
in time.
Sure, that gives me a higher order accuracy
right, the truncation error is much smaller,
the order of the truncation error is smaller,
convergence is better all of that is nice
but if you still have dissipation and dispersion,
you still have dissipation and dispersion,
what you may be; what the accuracy with which
you may be calculating something that thing
may not; may itself not be that accurate.
The other sources of errors that you have
to be careful right, so when you say unsteady;
computing unsteady flows, I want you to realize
first that what I am giving you, this is a
disclaimer basically and saying I am only
giving you a tiny introduction meaning that
there are a lot of serious issues involved
that which you have to struggle if you want
to get into doing actual unsteady flow.
It is not just that oh, I have a high resolution
scheme, I am going to take; I am going to
resolve timescales, I am going to resolve
length scales, I am going to take right, Runge
kutta in time and I am going to take a 4 points
centred or 4 point upwind or whatever in space
and I am going to get great answers right,
so you have to there is an issue here, right
that you have to worry about dispersion, dissipation,
there are other issues that you have to worry
about that.
You have to pay attention to it, okay given
that let me just go ahead in this naive fashion
saying that yes, I used Runge kutta in time
that is one possibility, one way to do it,
you can use Runge kutte on time, right but
then we have all this machinery that we have
just developed right now, right whether I
look at approximate factorization, whether
I look at you know, there is a whole host
of other schemes that I will talk about.
Or preconditioning the unsteady term, there
are other machinery that we have done that
the time that we are put into developing time
matching schemes to get steady state solutions,
am I making sense, we have spent efforts so
far developing schemes, so that we can get
steady state solutions to time matching schemes
fine, see I am going to; I am obviously, you
can see by the way and start going around
here, I am going to do something a little
fishy right.
So, I am trying to set you up for doing something
fishy, the other observation that we made
when we solve heat equation or if you want
to solve Laplace equation is; if you have
nabla squared phi equals 0, right you could
either solve this using sweeping in space
or you could actually add a term and look
for a steady state solution matching in time
okay, so you could either sweep in space or
match in time, fine.
So, since we did this and you could add any
term; unsteady term here, the question is
why do not I add an unsteady term to this,
why do not I just add an unsteady term to
this right, they say why would you want to
do it? Well, if I want to choose, if I; what
was the time step that I took in the demo
yesterday, what was the largest time step
that I took in the demo yesterday, do you
remember? 5 microseconds, you know what I
am saying that is it.
I want to take larger time steps, I want to
take something in milliseconds there, right
I want the spatial resolution but I want to
take time steps in milliseconds not in microseconds,
so then I would have to go to an implicit
scheme. So, if I went to an implicit scheme,
I have to struggle and solve a system of equations,
if I solve this using an implicit scheme right,
I have if I did an approximate factorization,
then I have lost the accuracy.
So, if I want to use, so if I want to use
for example backward space right, so if I
wanted to use backward time, so I could use;
so, let me see how does it go; 3Q pq + 1 – 4Q
pq; p is in space, q is in time, + Q pq – 1/
2delta t, right. I could go to a higher order
accuracy this way, am I making sense and do
an implicit scheme plus dou E/ dou x at q
+ 1 equals 0, right, I have discretized, I
have shown only this being discretized.
Am I making sense and I can do the same thing,
chain rule I can have a flux Jacobian here,
I can write a Delta form and so on, the only
difference is that now I have higher order
accuracy and in order to retain this higher
order accuracy but still take that large time
step, I will actually have to the system of
equations okay. I actually have to solve the
system of equations, am I making sense.
In this case because it is one dimensional
flow, fortunately I still get a tridiagonal
system, it is not that bad but the minute
you go to 2 dimensions or 3 dimensions it
becomes expensive, it is one thing, in the
demo I took 1000 or 100 points, right so if
you at it as a block matrix that is a 1000/
1000, 999/ 999 but 1000/ 1000 block it is
a 1000/ 1000 matrix or if you want to look
at it component wise, that is a 3000/3000
matrix.
And I am only solving a one dimensional flow,
am I making sense, it is an expensive process,
so we have already said how matching is the
same as sweeping, so we say why do not I add
an extra time, why do not I add an unsteady
term right, so the confusion comes the sort
of physical look, everybody's face comes saying
wait a minute, there is already a time, so
I add one more time, I create a pseudo time,
okay.
So, I add dou Q dou tau + that so, all I have
to do is; I have to discretize this dou Q
dou tau, in fact as I said this need not even
be Q it just has to be something that depends
on Q, so I will just say Q bar, is that fine
okay. Now, if you are willing to squint a
little and ignore the fact that this is time
and treat this tau; this pseudo time okay,
so terms that you will see is pseudo time
or you sort of admit that there are 2 times
or they are called dual times, right.
This is called either dual time stepping or
pseudo time stepping, if you are going to
go out and look at; look to see what are;
if you are going to go out and search right,
these are 2 possible search terms that you
would use, either a pseudo time stepping or
dual time step. Now, what I propose is; I
have this equation as I said we keep squinting
at this and treat that as though, it is not
time, I am going to match, I am going to use
a time matching scheme and tau.
And I am going to converge, go to convergence
and tau and when I go to reach convergence
in tau, dou Q dou tau will be 0 and I would
have solved the resulting system of equations
that is the plan okay, is that fine. So, I
would have; I am going to match; I am going
to do time matching in tau, I will use a higher
order accuracy representation for dou Q dou
t for the real time derivative.
This will evolve; it will evolve in the pseudo
time and when it reaches a steady state in
pseudo time, this will go away and I would
have effectively ended up solving the equation
that I want that I set out to solve, is that
okay and because I have a new coordinate,
so what is the price that I paid? I have a
new coordinate originally, one spatial dimension,
so we call it 1D but actually it is 2 dimensional
because I have x and t, now it is become 3
dimensional that is the price that I paid.
What is the advantage that I get? I am thinking
time matching, oh, I have done this before
I can do this, I can handle this okay. So,
as a consequence when I discretize this, I
get a Q bar pq + 1 and then I get an r + 1,
am I making sense, maybe I will write this
out separately, why do not I write it out
here separately so that you will get. Shall
I first do it with a wave equation, will you
be more comfortable if I do it with wave equation
first.
I think okay, maybe why do not I do it with
the wave equation first, then you can, maybe
I should have started that off okay. Dou u
dou t + a dou u dou x equals 0, right, we
will do it with the wave equation first and
then apply it there if you want. so I am going
to add a dou u dou tau, so if I were to discretize
this, remember now I will have 3, right subscripts
and it is a total of 3 subscripts and superscripts.
So, this becomes u pq + 1 r + 1 – u pq +
1r/ delta tau, that is a time derivative in
tau plus, we have choices, I will make a particular
choice and later on, I point out to you that
I have made the choice, u pq + 1, so you have
to decide, this is at q + 1, this is the time
that we are going to do, so that is r, choices;
I will make a choice r, there are 3 of them;
- 4 u pq + u pq – 1/ 2 delta + a, how come
they do not have a third subscript?
They do not have a third subscript, right
I mean, I asked how come they do not have
a third because they do not have a third subscript,
there is no pseudo time, these are in real
time, okay let me write it out and maybe I
will explain that so, a up + 1q, we will do
central differences; - up – 1q/ 2 delta
x, so what we are doing is; I will write it
here, q -1, 1, q + 1, we are here and we want
to go to q + 1, you understand.
The tau is a coordinate that occurs between
this and this everything is known below that
q is known, q - 1 is known, q - 2 is known,
they are all known when you converge, when
r, qr becomes qr + 1, when this converges,
when this goes to 0, you would have the q
or u will become uq + 1, you understand, this
is something that is happening in this gap
okay, is that fine, these are known, these
quantities are known, okay.
So, we are set, we can know this is sort of
like it, this looks like very suspiciously
like an explicit method, right it looks like
everything at r is known, so you just take
all of this; this equals 0, you take all of
this to the right hand side, u pq + 1, r +
1 equals u pq + 1 r + delta tau times, the
equation that you are actually trying to solve,
am I making sense; plus, minus, minus; - delta
times delta tau times the residue, right delta
tau times the residue.
These are all known, these are all constants
for all, these iteration in r, these are all
constants, the only one that changes is this,
is that fine, you keep on; so what would be
a good first guess for this 0; what would
be a first guess for that? Use some explicit
scheme; use some standard explicit scheme
you understand or u pq, you would be a good
guess, this is a cheaper guess, right you
could choose.
I will be honest, this is what we needed,
right with the suggestion that everybody made
is what we did the very first time that we
did this, u pq would be the good guess then
we got a little bolder and say hey, I can
actually take whatever explicit thing that
I was doing earlier and use that as the first
time step right, it is an improvement, so
you could use; you can use even with all this
satisfying the stability condition and all
of that stuff, it is still a little better.
You could use that as the initial guess, calculate
the r, update the u, you keep repeating this
process till it converges, okay I am not going
to do it here but you can actually do the
stability analysis for this just like we did
for FTCS okay, there is a stability condition,
we are using an explicit scheme here in r,
we are doing forward time in r, you get r
+ 1 explicitly in terms of all the terms in
r, it is an explicit scheme.
It turns out there is an associated stability
condition, fine okay, are there any questions?
So, how do you decide on delta tau and delta
t that will actually come out of the stability
condition, there is a stability condition
on delta tau and delta t, see there are 2
issues, when you say how do you calculate
it, it is like saying how do I decide on delta
t in my computation? The delta t in your computation
will depend on other parameters.
What is the accuracy you are looking for,
how much you know there are; it depends on
other parameter but there is a stability condition,
there are constraints right, so question you
should ask is what is the constraints; are
there any constraints on delta tau? There
are constraints on delta tau, if you ask the
question, are there optimal values for delta
tau, so that you get the steady state, yes
there are optimal values for delta tau, right.
So, you have to figure out how to pick that
every time you introduce a parameter, it is
not obvious that just like SOR, we introduced
an omega, so you can ask the question, how
do I pick omega? Well, yeah that is an issue,
if you get a good omega, it will converge
fast right, it is just a parameter that we
introduced, it is not part of the problem.
So, one way to look at it is; look at it as
a problem, it is a difficulty saying that
why should I do this, why do I do this?
It is just a headache I have one more parameter
to determine, the other way to look at it,
it is an opportunity, you look around figure
out, if you can get a delta tau or a delta
tau/ delta t that ratio if you get it right,
we will get very rapid convergence okay. So,
our experience with this just as the initial
transients; the initial first few time steps
the number of delta tau time steps that you
take is large so, if the order of 1000, 500,
1000 of that order.
But after that is like down to 4 or 5, okay
convergence is very rapid after that it is
only the initial; we have always found that
initially, it takes some time for the and
after that subsequent time steps, it is of
the order of 4 or 5, okay convergence, so
but you take some; you have to implement it
try it out see what happens, fine. What is
the other possibility; I said I made a decision,
what is the other possibility?
Maybe this I do using the full equations okay,
what is the other possibility? So, I have
Q pq + 1 r; r + 1 - Q that was Q bar, is not
it? Delta tau + 3 Q pq + 1, so you could instead
of choosing r, you could choose r + 1, then
it sort of looks implicit like; so it is implicit
okay, - 4 Q pq r; no r + Q pq – 1/ 2 delta
t. What do we do for E now? E which happens
to be a function of Q is also a function of
Q bar.
And if I represent E at r + 1 is E at r +
dou E dou Q bar delta Q bar right, anyway
I have used forward time here, I am not using
such an accurate scheme there, so I am doing
the same thing there okay, we have done this
the only difference is that I am doing it
with respect to Q bar, so this is dou E dou
Q bar, so I should most probably call this
A bar, right I am trying to write it in the
delta form now that is basically what I am
trying to do, okay.
So, this gives me delta Q; + dot, dot, dot;
delta Q bar/ delta tau +; here, I want a delta
Q bar, what am I going to do? I will add and
subtract 3Q pq + 1r, I will add and subtract
that so I will get a 3/2; 3 delta Q bar, you
have to be bit careful with that maybe, I
will do that a little; let me just leave that
I will get that, this is Q, remember this
is not Q bar, 3Q well, I can add and subtract,
it does not matter.
3 delta Q + 3Q pq + 1 r - 4Q pq + Q p – 1/
2 delta t + A bar dou Q; dou/ dou x A bar
delta Q bar equals; what is on the right hand
side, I take this Er to the right hand side,
dou Er dou x -; is that fine, everyone okay
and as we did before, we will use, we can
relate delta Q to delta Q bar, right, so delta
Q, we can write this, we can substitute delta
Q, you can write that delta Q as some P bar
times delta Q bar, right, where P bar would
be dou Q dou Q bar, is that fine.
All I have done this, I am just using chain
rule, I am just want to see I have some Q
Bar for instance, this could be a rho UT right,
these variables could be rho UT, this could
be a rho, this is our standard conservative
variables; rho, rho u, rho ut, right we may
figure, we may find and actually it is a fact,
we may find that using rho ut or rho up or
something may be better for the pseudo time,
okay.
So, for various reasons I will point out one
possible reason for a various reasons, so
we choose to have this; pick this variable
that is going to go to 0, the dependent variable
we choose to pick it, right and if we pick
it, then of course I have a delta Q, which
I have to convert to a delta Q bar, I have
no choice, so I do it through chain rule,
I just basically perform a change of variables
only for that single term.
So, my delta form then turns out to be I +
delta tau/ delta t that is that I specifically
write it because we had a question about that
multiplying P bar; 3/2 delta tau/ delta tT
multiplying P bar + dou / dou x A bar acting
on delta Q bar equals - delta * r of Q bar
and what is r of Q bar? The residue; dou E
dou x +; however, you discretize this plus
that term 3Q pq + 1 or - 4Q pq + Q pq – 1,
am I 
making sense?
In fact, r; as far as I am concerned I do
not write it this way, I write it as dou E
dou x + dou Q dou t that is the residue, so
I look at this equation, it looks like a familiar
equation, I have it in delta form, the residue
decides when I am done and when my solvers
done and when the residue is 0, I am solving
this equation with dou Q dou t represented
second order accurate or if you want to do
something fancy, you know maybe you can step
in and do it.
You understand what I am saying, is that fine
okay, so now the little twist that we throw
in; the little twist that we throw in is;
we say wait a minute, we just looked at an
acceleration scheme, pre conditioning the
unsteady term, so you can precondition the
pseudo unsteady term, so you basically take
this 
and you can multiply that by a gamma or a
gamma bar if you want, since we are saying
Q bar fine, right.
This is as I said, this is a sampling but
this gets you very quickly to where we are,
so the kinds of things that one can do, you
can add a pseudo term; the pseudo term is
going to go to 0, when you are; you can pre
multiply it by some gamma, if you pre multiply
it by some gamma there will be a gamma that
shows up there that is basically what happens,
if you pre multiply this by gamma what will
happen is; this will become gamma that is
the only thing that will change.
The change is a small change, the change is
seemingly a small change, the idea is that
if you have a code that you developed, can
I make a small change to the code to make
it run faster rate, right to make my convergence
better, so we can pre multiply this pseudo
dual time stepping unsteady term by the gamma
precondition that term, so that this converges
faster to what looks like a steady state and
tau but it is in fact a transient.
And when you have done that what you have
done is; you would have gone from Q to Q +
1, it will take a one step, am I making sense
then you shift; whole machinery shift, restart
again, fine great. As I said here, please
remember so I am only talking in terms of
oh, I addressed only one issue how can I take
a very large time step? I wish I could do
implicit schemes without all that effort.
Well, it is implicit scheme, there is a certain
amount of effort but the effort is what we
have been doing so far, so hopefully it is
not that difficult, we are just using the
ideas that we had in our time matching schemes
so far, so which means that you can now try
doing approximate factorization here, you
can do whatever you want in tau here because
when this delta Q tilde goes to 0, r will
be 0 and you would have solved your unsteady
equation exactly and more accurately, right.
Does that make sense, so you can precondition,
you can do approximate factorization all the
games that we are playing earlier, you can
do the same things here, everything that we
have done so far, you can do the same things
here knowing that you are going to solve the
full unsteady equations ultimately, right
but you do all of that stuff, you have taken
one time step that is the point to remember,
do not lose sight of that.
You do all of that stuff, you have taken one
time step, then you have to repeat the process,
the only question is; is this less expensive
or more expensive than solving the full system
of equations okay, fine and of course, if
you are going to solve a full system of equations,
there is whole bunch of machinery to allow
you to help you solve systems of equations
very efficiently, so the competition is there,
the comparison right, the computation of ideas
is there.
Because there are; it is not as though people
have not tried to solve large systems of linear
equations right, then there are algorithms
there too do not get; do not let me, you do
not allow me to give you the impression that
oh, this is it and the other one is difficult
to know it, there are other possibilities,
I am just basically saying we have developed
certain schemes, certain skills using to get
steady state solutions using time matching
schemes.
They can also be used to solve for the unsteady
problem fine, okay. In the next class, what
we will do is; we will look at some other
acceleration schemes right, we will look at
another class of schemes to increase the;
improve the performance of your solvers, fine,
thank you.
