Rec 4 | 18.085 Computational Science and Engineering I, Fall 2008 Video Lecture

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so but I'm open for questions as always
on every aspect or comments on every
aspect of the course yeah
the MATLAB problem
yes ok
ok so so let's remember I better
remember what that MATLAB problem is so
it's solving the equation minus u double
prime plus V u prime equals some Delta
function halfway along ok just two and
and I chose kind of unlike bound
boundary conditions like those so that
this gets replaced by a matrix K over
Delta x squared and this term gets is V
times some Center difference I don't
know whether I should use C we called C
earlier in the book C was a circulant
this is a centered first difference over
Delta X equal ax a right hand side ok oh
oh sorry all that multiplies the the the
discrete solution you okay I didn't get
that very well for the camera but maybe
it's I'll say it again we have the the
second difference that corresponds to
minus u double Prime and V times the
first difference that correspond
the vu Prime and then the problem is
what happens so the the physical problem
is what happens is V gets larger V grows
what how does the solution change as V
increases so it's a very practical
problem the V measures the importance of
this convection that the velocity of the
fluid and then n is the which is well
maybe maybe Delta X is Delta X is 1 over
n plus 1 maybe so that we would have n
interior points and we know the boundary
values so as then yeah it's a so the
question is what how do we interpret
this increasing in i-s we expect more
and more accuracy right as n increases
Delta X getting smaller our idea is that
the this should be a better and better
approximation to the differential
equation the question about the
eigenvalues of this matrix I don't know
exactly the answer what how the so the
eigenvalues did you discover that the
eigenvalues suddenly they change their
real for small values of V n n but then
there's some point at which they become
complex and and a point that involves
both V and Delta X actually we end in
it's kind of interesting I guess at this
moment I interesting is the key word I
don't know what physical I don't know
really the so what is the physics going
on here so I have a
I have a a 1d flow and it's blocked by
these boundary conditions are sort of
walking it at both ends okay and I'm
looking for a steady state and I'm
feeding in this source here so I'm
feeding in a source halfway along so I
think that would be right now what's
going to happen the roughly the the this
term is I think flowing to the right so
it's going to carry suppose this was you
know smoke or particles or whatever it's
a this is like an environmental
differential equation or chemical
engineering equation just oh and more
more applications so I think of it is
this term is carrying the flow with it
like that way so I'm expecting your
solutions to be larger on this side and
how did how do we get any action at all
on the left that comes from the
diffusion the the the the particles when
they enter get carried away and faster
and faster as V increases but at the
same time they diffuse they they bounce
back and forth and some little some bit
of it goes this way but not a lot I'm
guessing that that the solution would
have a profile you know that would be
kind of small and then there's whatever
jump there has to be here and then
larger there oh but then I have to get
it to zero it's a it's a it's a strange
situation and maybe I take this chance
to mention it what happens as V gets
really really large the
you could say okay as as we gets good
gets extremely large forget that term
this is the important term vu prime
equals that which we could easily solve
but this equation with just this term is
only first order right it only has a
first derivative and how many boundary
conditions would we expect to have if I
gave you an if this wasn't here and I
gave you just a first-order equation one
and we've got two which we're imposing
so this sort of limit as V increases it
produces something called a boundary
layer the the solution it's forced to
satisfy these boundary conditions one of
them it's quite happy about the other
one it didn't really want to satisfy it
only satisfied them because this because
I had to originally with this part but
as this takes over the the struggle to
satisfy that second boundary condition
that it really doesn't want to satisfy
is resolved by something a layer that it
just makes a sort of exponential
correction at the boundary to get to
where it's supposed to go so anyway this
is a model problem that has boundary
layers those are terrifically important
in aero aerodynamics you know you have a
layers around the actual aircraft so
lots of physics and computational
sciences hiding in this type of example
well so those are a few words but not an
answer to your question which was I
guess I'm happy if you if you for
example let me know where the
eigenvalues changed from real to complex
I mean it happens but you can probably
if you look at the matrix you can see
what happens at the point where
that that change have change takes place
okay that some comets and with my with
the MATLAB pure MATLAB part I guess I'm
hoping and the graders are hoping that
by providing a code we'll get a pretty
systematic set of answers for the for
the requested V equal what was it three
and twelve maybe or something and they
requested ends Delta X's but so I'm
hoping that like everybody's graph is
going to look pretty similar for those
those requests and then if you could add
if you could do a little exploration
yourself and make that a fifth page or
really less than a page about take V
larger take n larger what what happens
I'm very happy or you know like or any
direction I mean just think of it as a
mini project to do the requested ones
and then to do a little experimentation
yeah yeah and and I'll just be
interested with any about any anything
you observe right so so there's no right
answer to that to that part okay is that
any help with a MATLAB so let me just
say since you showed up today if you
have trouble with a MATLAB and you need
Peters help at noon Friday it would be
okay to turn into MATLAB later on Friday
I shouldn't say that but I just did okay
so there there you are right so yeah so
I mean this is this course as you've
begun to see
you know I know that you have lots of
demands on your time and at some point
sometimes you have funeral job
interviews all sorts of stuff
conferences that you have to go to we
deal with that so just give me the
homeworks as soon as you can if you're
ready Friday at in class time that's
perfect if you are stuck on MATLAB and
you want Peters just to go to Peters
discussion here which is follows the
Friday class in here that's fine too
okay so that's some thoughts about the
MATLAB and and it's that part of it
partly open-ended okay what about lots
of other things in this course homework
came in but what how was the homework
maybe I just take this chance to ask was
it a reasonable length the homework
three they're probably the pset oh not
too bad okay yeah yeah okay and I'll I
will be able to post solutions because
several people are contributing typed
solutions that that will there can be
that could go on on the math at mit.edu
slash CSE website yeah or they could
also go on our eighteen oh eight five
website for this semester
good okay help me out with some
questions thanks
okay yeah that was fast and of course
Wilson so the question is about element
matrices in lecture eight I guess it was
where we where we were we were
assembling this I use the word that's
used in finite element by finite element
people we were assembling K and one way
to do it was to multiply the three
matrices but that's not how it's
actually done at least yeah it's it's
actually assembled out of small matrice
maybe small K would be there the right
so this would be a K element out of out
of smaller element matrices okay so for
example what's the element matrix for a
safe one for each spring in in this
particular problem of Springs and masses
so so let me draw some Springs some
masses some Springs another mass more
Springs fix not fixed whatever may be
fixed free the way I've drawn it there
so here's a Springs with spring constant
c2 and we could look at the contribution
to the whole matrix coming from this
piece of the problem this was actually
the the finite element method has a like
a wonderful history of people and it had
different names way back of people
seeing the structure as broken in pieces
and then connected together and and then
what did a typical piece look like so a
typical piece there well you know let me
just write down what this matrix is
going to be the little element matrix is
from
coming from spring - so this would be
like element two will be it'll have a
situ outside yeah up with the situ
outside and then inside will be a little
this guy okay and we we can talk more
about why it's that one okay but just to
have the element matrix there on the
board okay so my claim is that this
piece of the this is a small piece of
the big K so the big K matrix what's
what's the size of the big K of K itself
3x3 in this case yeah three masses so
it'll be three by three okay so I'm
thinking that this spring this spring
which connects mass one to mass - so
it's it's it's only going to be like a
two by two piece a little local piece
you could say that that little K fits in
this is assembled into that I better
call that K - right k - so it's a little
K that sits up in this two by two block
and and doesn't contribute to the rest
okay then let's just draw the rest of
the picture so this would produce an
element matrix that looks just the same
that's the beauty of this that that all
the elements apart from change in the
spring constant look the same so
there'll be a little CC a little a k3
and where will it go
this is the whole part of the point
it'll go to the lower right down here
overlapping overlapping because this
mass is part of it is to attach of that
spring and to that one so these little
element matrices they overlap and and
you just need if you can imagine
the the the code that's going to do this
is you you need a list of Springs and a
list of masses and a list of the
connections between them
and it'll sit in here because it's 2 by
2 so that's 2 by 2 that's 2 by 2 and and
in this over this overlap entry will be
a c2 from the upper box and a seat it'll
be a C 2 plus C 3 as we discovered by
direct multiplication so that's that
spring now what about this first spring
so there's a little k1 now k1 is should
be you look the same as this except what
so is it gonna be a difference with the
with this spring because because it's
it's it's because of this fixed in so
you know there's no mass zero you know
it k1 would normally sit up here but
actually there's only it's only it's
only going to be one by one it's k1 well
so the k1 little element matrix would be
well it would look like C 1 1 minus 1
minus 1 1 and then the boundary
conditions knock those out and an
interesting point
if you're writing big code you have to
decide do I as I create k1 this little
element matrix do I pay attention to the
boundary conditions and then k1 would
just be that that single number C 1
which would sit there so so up there
will be C 1 from the k1 matrix and C 2
from the k2 matrix that's the entry
that's the entry there and then as I say
encoding finite elements which you guys
may do at some point
there's a question what's efficient
shall I pay attention to the boundary in
creating these element matrices or shall
we do it later and
rule seems to be do it later so what
gets created in finite elements is a is
is in our language it would be a free
free matrix it's a matrix with with
where even this spring has a two by two
piece and then so so what's the problem
with the free free matrix the free free
free matrices those with no supports
those matrices will be singular right
singular because the the vector of all
ones if you multiply that K the the free
free matrix times the vector of all ones
you get zero I mean the whole thing
could shift and we'll have other rigid
motions in two dimensions so yeah let's
just think ahead here what are the rigid
motions the null space of K you could
say in a two dimensional for a two
dimensional truss so the the whole I've
got a bunch of boards and Springs think
of a mattress okay I'm in two dimensions
a mattress as a bunch of Springs
connected in a 2d grid what what do i
and and say say it's free it at both
ends so what can I do grow a mattress oh
my gosh I didn't expect that to be on
videotape but what so it's in the plane
we're in 2d here what can I do with if
there are no no boundary conditions well
I can shift it to the right right that's
our one one one one one
what else can I do I can rotate and I
can shift it the other way so there
would be three rigid motions for the 2d
problem two translations and a rotation
and when you get up to three dimensions
do you want to guess how many what's the
number and 3d six number six
three translations and rotations around
three axes so those either one rigid
motion or three rigid motions or six
rigid motions have to get the boundary
conditions eventually have to remove
those have to have to knock out rows and
columns and remove them but my point was
just that quite efficient to do it later
you know the picture is so clear here so
so the actual matrix would be four by
four the unreduced matrix and then and
then when we fix node zero there that
would knock out that part and leave the
three by three that we want so this is
just discussion of how to think of this
matrix K so the direct way to think of
it was as was as a product of big
matrices but in reality it's assembled
from these element pieces and of course
our goal in talking about finite
elements will be to see that it'll come
up again of course yeah okay so that
your good question led me there but did
I answer the question okay so that's a
thought about and maybe the way I
mentioned it in class was to notice that
these guys these element matrices can be
thought of this way yeah it's made it is
matrix multiplication but done
differently it's a column of this matrix
times the number see here times a row of
this so so it's matrix multiplication
columns times rows so you can multiply a
B columns times columns of a times rows
of B and and then add add over from 1 to
n over
so column of a times row of B so a a
column man is a vector like this a row
is a vector like that I'm yeah and so
this is and the result is a full size
matrix but if the column is concentrated
at a couple of nodes and the row is then
it will have zeros everywhere except at
that so this is like this is the element
matrix with the C included that would be
the element matrix already blown up to
full size by by adding zeros elsewhere
so I mean the heart of the the heart of
the element matrix is is is where the
action is on that spring and then when
we assemble it of course it it doesn't
contribute down there yeah so it's the
technology I find that elements it's
just it's quite interesting and it fits
it's it's a beautiful way to see the
discrete problem okay
ready for another question of any sort
so thank you good yes
oh yeah yeah sorry it got under the
problem set and then I thought so this
is let me write those words down first
singular value decomposition
well I won't write all those words that
would that's the only it's a wonderful
thing great it's a highlight of matrix
theory except for the length of its name
so SVD is what everybody calls it yeah
yeah it's only you know like every year
now people appreciate more and more the
importance of this SVD the singular
value decomposition so shall I say a few
words about it now it's just a few it's
yeah so yeah so it's section 1.7 of the
book and my ID my thought was hey we've
we've done so much matrix theory let's
including eigenvalues and positive
definiteness let's get on and use it and
then I can come back to the SVD in a
sort of lighter moment because I'm not
thinking you will be responsible for the
SVD eigen values I hope you're
understanding
positive definite I hope you're
understanding you know that's that's
part of the course stuff but SVD is a
key idea in linear algebra but we can't
do everything okay but we can say what
it is okay so what what what's the first
point the first point is that every
matrix even a rectangular matrix has got
a singular value decomposition so the
matrix a can be M by n it wouldn't I
wouldn't speak about the eigenvalues of
a rectangular matrix you know because if
I multiply it you remember the
eigenvalue equation
wouldn't make sense ax equal lambda X is
no good if a is rectangular right the
input would be of length N and the
output would be of length M and I could
never this wouldn't work so so I ghen
vectors are not what I what I'm after
but somehow the goal of eigenvectors was
to diagonalize the goal of eigenvectors
was to find these special directions in
which the a matrix a just like acted
like a number and then as we saw today
in part that that's partly still up here
we could solve equations by eigen
vectors by looking for these following
these special guys okay what do we do
for a rectangular matrix what replaces
this so this is now not good the idea is
simply we need two sets of vectors we
need some V's and some use okay so
that's the that's the central equation
of the SVD now what can we get so now we
have more freedom because we're getting
a bunch of V's that have these these
guys are in RN they have length and
write to multiply m8 a times V and the
output is is so the n of these envies
they're in we're in n dimensional space
I'm so those are called singular vectors
they're called write singular vectors
because they're sitting to the right of
a and these guys these outputs will be
in these RVs in RN and dimensions these
I have M of these M use in M dimensional
space and these things are numbers of
course and actually there
they're all greater equal zero so we can
get by allowing ourselves the freedom of
to singular write singular vectors and
left singular vectors we can get a lot
more and we can get here's here's the
punchline we can get the V's to be
orthogonal worse than normal just like
the eigenvectors of a symmetric matrix
we can get these V's these singular
vectors to be perpendicular to each
other and we can get the use to be
perpendicular to each other what we
can't what we're not shooting for is the
V's to be the same as the use that that
they're not even in the same space now
if if the matrix a is rectangular and
even if the matrix a is square but not
symmetric we wouldn't get this
perpendicularity but we get it in the
SVD by having different V's and use yeah
yeah here's my picture of linear algebra
I don't so this is the this is the big
picture of linear algebra we have the we
have all we this is this over here is n
dimensional space we have V's okay
think of that as n-dimensional space
it's kind of a puny picture but when I
multiply by a I take a vector here I
multiply by a so let me just do that for
this is I multiply by a I take a vector
V well I already put V I take a V and I
get something over here and this will be
the space of use now okay now I have to
ask you one thing about linear algebra
that I keep hammering away if I take any
vector and multiply by a
what sort of what what do I get if I
take any vector V like one two and I
here's here's a a say three four five
six seven eight so that's a times V what
can you tell me about a times V that
goes a little deeper than just telling
me the numbers it's a linear combination
of the columns so this is what I'll the
these are the outputs the Avs so this
space is called the column space it's
all combinations of the columns these
these are all combinations of the
columns that's the column space it's a
bunch of vectors and so the point is
that these use are that I have that like
a fantastic choice of axes linear
algebra person would use the word basis
but geometrically I'm saying they're
fantastic axes in these spaces so that
if I choose the right axes then in the
two spaces then one axis will go to that
one when I multiply by a the next one
will go to that one the third will go to
that one
it just could not be better and the part
that and that's the statement okay why
no maybe I'll just say a word about Y
where it's used or Y why it's useful in
Owens in so many applications where
where do we get yeah well most of you
are not biologists and I'm not certainly
but we are we know that there's a lot of
interesting math these days and a lot of
interesting experiments with
say jeans so what happens so so we've
got about 40 people here and we take we
measure say the well we get data that's
what I'm really gonna say somehow we get
a giant amount of data right probably 40
columns everybody here is entitled to
its column to be a column of the matrix
okay and and the entries will be like
you know have you got TB what height all
this stuff right okay so so you got an
enormous amount of data and the question
is what's important in that data you
know you've got say a million entries in
in a giant matrix and you want to
extract the important thing well it's
the SVD that does it you take that giant
matrix of data you find the V's and the
Sigma's and the use and then the largest
Sigma's are the most important
information in in the if things are
scaled and statistics is coming into
here into this also I could give a
sensible you know more much more
mathematical discussion of one word one
name it goes under is principal
component analysis so that's that's a
standard tool for for statisticians
looking at giant amounts of data is to
find the principal components and those
will be come from these eigen vectors
okay well your question about the SVD
set off that discussion I only add a
couple of words these V's and these
views are actually eigenvectors but
they're not eigenvectors of a they're
these are happen to be the eigen vectors
of a transpose a and the use happen to
be eigenvectors of a a transpose and the
just the the linear algebra comes
together to give you this key equation
and and of course you and I know that if
I'm looking at the eigen vectors V of a
transpose a they a transpose a is a
symmetric matrix in fact positive
definite so that the eigenvectors will
be orthogonal the eigen values will be
positive and then this one is coming
from the eigenvectors of a a transpose
which is different right yeah so if the
matrix a happened to be square happen to
be symmetric happen to be positive
definite this would just be ax equal
lambda X I didn't have to cross it out
so if it was I'll use K for that right
so if the matrix a was one of our
favorite matrices was a K that would be
the case in which the SVD is no
different from eigen values the V's are
the same as the use the Sigma's are the
same as the lambdas all fine but so
we're this is sort of the way to get the
beauty of positive definite symmetric
matrices when the matrix itself that you
start with isn't even square like like
this one like that okay more than I
wanted to say more than you wanted to
hear about the SVD anyway that's what
else
yes thank you
yes more about sure okay let me repeat
the question so this was refers to this
morning's lecture lecture nine about
time-dependent problems and I and the
point was that when I have a
differential equation in time there are
lots of ways to replace it by difference
equations okay and one big so let's take
the equation D u DT let's make it
first-order is some function of you
often NT that would be a first-order
yeah it's good to look at just when when
I write that much down what have I got
I've got a first-order differential
equation right is it linear no I I'm
allowing something like this function of
you could be sine of U it could be U to
the 10th power
it could be e to the U so this so that
this is how equations really come you
know the linear ones are the model
problems that we can solve this is how
linear equations really come ok now I
could think of Euler thought of let's
give away their credit here so forward
Euler forward Euler and backward oil and
the point will be that this guy is
explicit so can I write that word so I
don't forget to write it explicit and
that this guy will be implicit and this
is a big a big distinction and and it
will see it alright so this says u n
plus 1 minus u and over delta T is the
value of the slope at the start of the
step so that's a weather that's the most
important most basic first thing you
would think of it replaces this by this
you you start with you 0 and from this
you get you one and then you know you 1
and from this you get you 2 and the
point is at every step you just this
equation is telling you what you one is
from you zero you know you just move you
zero over to that side
you've only got stuff you know and then
you know you want
contrast that with backward Euler
so that's u n plus 1 minus u n over
delta T is f now what am I going to put
there I'm gonna put the end the point we
don't know yet so is the slope at U n
plus 1 and the time that goes with it T
n plus 1 that would be T n is just a
shorthand a shorthand for n delta T you
know n steps forward in time got us to
TM this is one more step now why is this
called implicit because how do I find u
n plus 1 out of this I've kind of solved
for it it's much more expensive because
there will be a probably a nonlinear
system of equations to solve will take a
little time on that but of course
there's one outstanding method to solve
a system of nonlinear equations and
that's called Newton's method so Newton
is come in Newton's method is this sort
of the first yeah the really the good
way if you can if you can do it to solve
well when I say solve I mean set up
iteration which f
maybe three three loops or five loops
will be accurate to enough digits that
you can say okay
that's un plus one on to the next step
then the next step will have the same
equation but within up one so it would
be un plus two minus UN plus 1 so you
see the extra work here but it's more
stable this the way this one spiraled
out this one spiraled in in the in the
model problem I hope you look at that
section 2.2 which was about the model
problem that that we discussed today
there's more to say I mean this is the
central problem of time dependent
evolving initial value problems your
this is your sort of marching forward
but here each step of the march takes an
inner loop which has to work hard to
solve to find the where you march to
yeah is that a help to indicate what
because that this is a very fundamental
difference right then as I said and
Chapter six of the book develops
higher-order methods these are both
first order first order accurate the
error that you're making is of the order
of delta T that's not great right
because you would have to take many many
small steps to have a reasonable error
but so these higher order methods allow
you to get to a great answer with bigger
steps yeah yeah a lot of thinking has
gone into that and it's somehow it's a
pretty basic problem just to mention say
for matlab OD e45 yes maybe the work or
code to solve this type of a problem and
it's the method is not Euler that would
not be good enough it's called runner
Kutta two guys one could have figured
out a formula that got up to a fourth
order accurate yeah so it's just that's
the that's the thing and then if we
looked further about this subject we
would we would distinguish some
equations that are called stiff so I'll
just write that word down some equations
the iteration is particularly difficult
to you have two time scales maybe you've
got things happening on a slow scale and
a high scale multi scale computations
that's where the subject is now and so
there would be separate codes with an S
in their names suitable for for these
tougher problems stiff equations okay
that's a I guess one thing you sometimes
get on this in the review session is a
you know look outside the scope of what
I can cover and ask homework problems
about okay I'm sure hoping that you're
assembling the elements of computational
science here you know yeah first what
are the questions what are some answer
is what are the do the issues what what
what do you have to balance to to make a
good decision okay another time for
another question if we like if you yeah
thank you
stability yeah right correct I will tell
you that yes right this here okay so
let's see first if we want a matrix then
this has to be a vector of unknown so I
I'm thinking now of a system of this is
n equations I've got you as a vector
with n components I've got n equations
here the notation I can just I can put a
little arrow over it just to remind
myself okay and if I want to get to a
matrix I better do the linear case right
so I'll do the linear case okay yeah so
yeah so if I the the big picture is that
the new values come from the old values
some there has to be a matrix
multiplying anytime we have a linear
process so that you know so that new
values come from old values by it by
some linear map a matrix is doing it and
if we right so there's a sort of growth
matrix can I just put down some letters
here I won't be able to give you a
complete answer but this will do it okay
so so UN plus 1 is some matrix times you
in that's that's the way that's what I
wrote down this morning for a special
case it's got to look like that for a
linear problem so now and let's suppose
that we have this nice situation as
today where it's the same G at every
step so the problem is not changing in
time it's it's not it's not it's it's
linear it's all good what is the
solution F
n time steps compared to the start so
give me a simple formula for the
solution to this step step step equation
if I started at initial value u zero and
I look to see well what matrix gets me
over n steps would Y right there G to
the N right G to the N ok so now comes
the question let's see the stability
question is whether do the powers of G
to the enth grow and here's the point
that the continuous problem that this
came from it's got its own growth or
decay or oscillation this guy the
discrete one has got its they're going
to be closed for a step or two for one
or two steps I'm expecting G - I'm
expecting that this is a reasonable
consistent approximation to my problem
but after I take a thousand steps this
one could still be closed or it could
have exploded and that's the stability
thing that the the choice the choice of
the difference method can be stable or
not
and the only it's V is going to be the
eigenvalues of G that are the best guide
yeah so in the end people compute the
eigenvalues of the growth matrix and
look to see are they bigger than 1 or
not
okay let's just let's close with that
example cuz that's a that's a good
example so it it fits this but not quite
because it wasn't completely forward or
completely backward and let's just write
down what it was in that model problem
and find the matrix okay so can can you
remind me what I wrote today so you was
it you first you n plus one was u n plus
delta T V n right and then the new
velocity was the old velocity and it
should be minus delta T times the you
say because our equation is what if
these are representing the equation u
prime equal V is the first one V prime
equal minus U is my second equation so
and then the point was I could take that
because I know it already I could take
it there right okay good
where is the matrix here okay could you
create I want to find the growth matrix
that that tells me now my growth matrix
of course U is standing this is U n V N
and this is U n plus 1 VN plus 1 and
this is a 2 by 2 matrix and I hope you
can we can read it off or find it anyway
because that's the key alright let's
let's get how could we how can we get a
matrix out of these two equations
let me just ask you to look at them and
think what what are we going to do
that's a that's a good question what
what shall I do I would like to know the
new use UV from the old what will I do
bring stuff on to a UN plus one on its
side and in on its side okay so that
just means then that I can I just this
guy I want to get over here can I do
that with it with my wither racing okay
so it's going to come over with a plus
sign okay so I guess I see here an
implicit matrix acting on the left and
an explicit matrix acting on the right
so I see a unvn and I see my explicit
matrix shall I call it e that's the
right sides I see a 1 and a 1 and a
delta T and now my left side of my
equation the the n plus 1 stuff what's
my implicit matrix on the left sides of
the equation well a 1 and a 1 and plus
delta T is here right and it's called
that I for implicit or yeah okay are we
close I think that looks pretty good
what's G what's the matrix G now you
just have to have a like oh we begin to
develop a little faith and yeah I can
deal with matrices
I couldn't move them around get them
there under my control so I want I
wanted this picture I want the I want to
move everything to the right what what
mate what do I do to move everything to
the right-hand side to move I want that
to be over there it's an inverse it's
the inverse so G
the matrix G there is I bring I over
here it's the inverse of I times the e
and I can figure out what that matrix is
and I can find its eigenvalues and it'll
be interesting yeah yeah
so that actually that's the perfect
problem I mean if you want to see what's
going on well the book will be a good
help I think but that's the that's the
growth matrix for leapfrog and I'll tell
you the result the eigenvalues are right
of magnitude one as you hope up to a
certain value of delta T and then when
delta T passes that stability limit the
eigenvalues take off so that this method
is great provided you don't you're not
too ambitious and take too large a delta
T yeah yeah okay thanks for that last
question thanks for coming