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PROFESSOR: OK.
So I'll start now with the third lecture.
And we're into partial differential equations.
So we had two on ordinary differential equations,
and the idea of stability came up, and the order of accuracy
came up.
And those will be the key questions today --
accuracy and stability.
We do get beyond that, actually.
I mean, to look at the contour, at the output from a difference
method.
Once you've decided, yes, it's stable, yes it's accurate,
you still have to look at the output
and see, you know, how good is it, where is it weak,
where is it on target.
But first come the accuracy and stability questions.
And beginning with this equation.
So this will be the simplest initial value
problem we can think of.
I call it a one-way wave equation.
Notice the two-way wave equation, which we'll get too,
would have second derivatives there.
And that will send waves in both directions.
This, we'll see, sends a wave only in one direction,
so it's a nice scalar problem.
First-order, constant coefficient,
that's the velocity.
Perfect.
The model problem for wave equations.
Because it's just first order, I just
begin it with the function at time 0.
So I'm looking for the solution at time t to this equation.
OK.
So that will not be hard to find.
And I want to do it first for pure exponentials.
When u of x_0, the initial function,
is a pure exponential e to the i*k*x.
Because it really pays, to -- Fourier is always around.
And to see what happens, suppose this is given, for example --
so this will be my example, e to the i*k*x.
OK.
Now what's special about e to the i*k*x?
The special thing is that with constant coefficients and no
boundaries, the solution will be a multiple of e to the i*k*x.
In other words, we can separate variables.
Maybe I'll put it here.
I can then look for the solution u to be a function times --
a number really, but it will depend on the time,
on the frequency k, and the time t -- times e to the i*k*x.
You see how that separated variables?
x is separated from t, and the frequency controls what
this growth factor will be.
So this growth factors is a key quantity to compute.
And we compute it just by substituting
that hope for a solution into the equation.
And getting an equation for G, because e to the i*k*x will
cancel.
That's the key point, that everything
remains a pure harmonic with that single frequency k.
So if I plug it in, I get dG/dt --
I'm taking the time derivative of u -- times e to the i*k*x,
right?
That's du/dt.
Because t was separate from x, that's what
I get when I plug in this u.
So I'm always plugging in this u.
What do I get on the right-hand side? c, the x derivative.
So that's just G. The x derivative will bring down i*k,
e to the i*k*x.
No surprise that I can cancel e to the i*k*x, which is never 0.
And I have a simple equation for G. Linear,
constant coefficients, but the coefficient
depends on k, of course.
The coefficient is i*c*k, and of course the solution is G is
equal to -- we're back to our simplest model of an ordinary
differential equation.
Now the coefficient a is now i*k*c,
so it's e to the i*k*c*t, right?
That shows us what happens to the e to the i*k*x.
So now I'll put that here, because now we know it.
So we have altogether e to the i*k*c*t, and here's an x.
So let's put those together as x plus c*t,
because that's a guide to what's coming.
So that's the solution: separated variables, looked
for the growth factor, tried an exponential.
And we'll do exactly the same thing for difference method.
So this was von Neumann's brilliant idea,
watch exponentials, watch every frequency,
and see what happens to the multiple of e to the i*k*x.
Now here is very special, because all frequencies are
producing this combination x plus c*t.
So when we combine frequencies -- that's what Fourier said,
take combinations of these things --
the solution will be a combination of these things.
And what do we get?
If our original function was u of x and 0,
it's a combination of these, e to the i*k*x's.
And at later time, we have the same combination,
by linearity of these guys.
All that's happening is x is getting changed to x plus c*t.
So, I'm getting the answer.
Here was the answer for one exponential.
But now I'm going to do the answer for a general u,
and it's going to be easy to pick off.
It's just like picking off this x plus c*t.
It will be u at x plus c*t at time 0.
That's the solution.
For every u, not just for the exponentials,
but for all combinations of them,
which give us any u of x and 0.
So u of x and 0 was a combination of e
to the i*k*x's, and then the solution is the same
combination of these guys.
Well, we can understand what that solution is.
It's what I call a one-way wave.
Let me understand what's happening here.
So here's the key picture now.
I want to a draw a picture in the xt-plane.
So I'm not graphing u here.
This is a picture in the xt-plane that will show us
what this formula means.
Because this is the solution that we then
want to approximate by difference method.
OK.
So what does this mean?
This means that the value, for example, u of 0, 0 --
so there's the point (0, 0).
Here's what's happening, at a later time, say up here,
some later time, the value of u --
this is the line x plus ct equals 0;
x plus ct equals 0 along that line.
And what's the point?
On that line, the solution is the same all the way along.
So whatever the initial value is here,
it travels, so that u at this point is u at this point.
So let me to call that point P, and say what I mean now.
I mean that u of P is the same as u at 0, 0.
Because x plus c*t is the same here and here.
Let me take another point, let's say x_0, another initial value.
Then, again, along the line like this --
this will be the line x plus c*t equals capital X,
because it's the line where x plus c*t is a constant,
and the constant is chosen so that it starts at this point,
at t equals 0, X is big X. So now -- let's see --
I better make clear -- that this line carries this initial
value.
This line carries this initial value.
You see it?
The value of u all along this line
is the value it started with right there.
And you know the name for those lines?
So these are lines, they happen to be straight here
because we got constant coefficients, straight
and even parallel.
And they're the lines on which the information travels.
Whatever the information was at times 0.
By information, I mean the value of u
at 0, 0, or the value of u at this point.
That information travels along that line
to give this solution all along the line.
And the name of the line, anybody know it?
Characteristic.
So this is a characteristic line, they all are.
This too.
All these line are characteristic lines.
Can you read characteristic?
I should of have written it a little better.
So this is like a key feature of wave equations.
It's a key feature of wave equations
that we will not see for heat equations.
And so let me just say it again, because it will bear also
on the stability question for difference equations.
The information, the true information,
for the true solution, travels along characteristic lines.
We're in one dimension here, which
is a major simplification.
In two dimensions we'll have characteristic cones.
In three dimension certainly we'd call it a cone.
And actually, I guess what I'm saying
is that if I speak a word, you know,
my voice or the sound waves solve the 3D wave equation.
So when I snap my fingers, that sound
travels from this point out along characteristics.
And of course it travels in all directions,
so we've got a more interesting, more complicated picture
in higher dimensions, but we really
see it here in one dimension.
OK.
Now, I should also include a graph of the solution.
What does the solution look like?
Maybe I'll find a spot here for that graph.
Now I'm going to graph u.
This is still x.
So I'll graph u at maybe two different times.
So the starting time.
Suppose we have a wall of water.
So this will be u of x and 0, a step function, maybe at 0.
So u of x and 0 is 1 and then 0.
OK.
So there's a typical starting initial value.
Well, I don't know if it's typical.
Pretty special to have 1 and then 0,
a perfect step function.
But this will illustrate --
Now I want to ask, what's the solution to the equation?
What is u of x and t?
I want to graph u of x and t.
What what happens?
You have to assemble, sort of, this picture, which
is in the xt-plane, into a graph here of u of x and t.
What happens?
Well all these zero starting values travel along
characteristics, and tell us that u is 0 along all
these characteristics.
So along all these characteristics, u is 0.
This here, the starting value is 1.
So along these characteristics we're starting with a 1,
we end with a 1.
Do you see what happens?
How do I graph it?
The wave is moving.
The wall of water is moving that way, moving to the left.
Because I'm taking c to be a positive number.
So u of x and t is a wall, or a wave you could say,
moving left with speed c.
Every solution to the equation is doing this.
And in this particular case it's a wall of water
and it stays a wall of water.
That the point.
Shapes are not changed as we go.
There's no -- dispersion would be the right word, I think.
We would expect, in a general equation,
that the shape of the wave would change a little as it travels.
In this equation it doesn't change.
The shape stays the same.
See, again I'm thinking of this wave as a combination
of pure exponentials.
And they all travel with the same speed c,
so the whole wave travels with speed c.
And at a later time, that's the graph of the solution.
OK.
So that much is all we have to do, all we can do really,
in solving the differential equations.
It's a simple model, certainly.
But now I come to the difference equation.
Well, when I say the difference equation,
I should say equations plural.
Because here I've written four possibilities.
Three of them are actually used, and one is a disaster.
But none of the four is perfect.
Most people would say Lax-Wendroff
is the best of these four, because it has
one extra order of accuracy.
Oh yeah, yeah.
I guess these all have only first-order accuracy.
But Lax-Wendroff moves up to second.
So that's a method that everybody knows,
because it gets up to second.
But actually upwind is the first one to start with.
Let me start with that one.
OK.
So I've multiplied up by the delta t that should be
in the denominator there, especially so that you would
see -- so upwind, or forward difference --
I'm taking the standard forward difference in time --
all these methods are going to be explicit,
so they're all going to be solvable in a simple way for u
at the new time.
So a forward difference in time for du/dt,
and c times a forward difference in space for du/dx.
So that's the first idea that would occur to you,
and it's pretty reasonable.
We can check its accuracy, but you can guess what it would be.
What's your guess on the accuracy of this method?
You're seeing a forward difference
in time, a forward difference in space.
The odds are those finite differences give you
an accuracy p -- if we use p as the measure of accuracy --
which would be probably 1, first-order accurate.
So we'll see the accuracy as only p equal to 1.
OK.
What about the stability?
I'm looking at this first method.
So one comment about the method is
that this ratio c delta t over delta x,
let me call it r, that's a key number.
it's called the Courant number by some.
So in the literature you might see the phrase Courant
number for that ratio r.
And maybe my first point is, stability
is going to put a bound on r.
We can tell right away, for this explicit method
and for any explicit method, that r can't be unlimited.
There's going to be a bound on r, which
it means a bound on delta t.
It's telling us that delta t has to be less than some constant
times delta x.
And stability is the job find that constant.
How small does delta t have to be?
Now let me sort of understand this forward difference method
in this picture.
So I'm going to do the forward difference, delta t of u
equals that ratio times the forward difference of the space
difference, so time differences is that ratio times this space
difference.
OK.
Method 1.
OK.
So I guess somehow, you want to see that the --
you want to see a picture like this for the difference
equation, So the difference equation,
we imagine we've got steps delta x, all these steps are delta x.
Let's suppose that's 0.
OK.
So what happening now?
And time going this way and I'm solving this equation.
What's happening here?
Let me copy this method, u at a position j --
so this might be the point j delta x --
at time n plus 1 is -- I want to write this equation, as always,
in the most convenient way.
So what am I going to do?
I'm going to take this u of x and t
and put it on the right-hand side.
So then my equation looks like that ratio r times u
at j plus 1 at time n.
That's this.
I'm one position over, but I'm at the given time.
And then what's the coefficient of u_(j, n)?
So can you just tell me, what do I put in there?
So this will be that nice way to look at our difference
equation.
Each new value is a combination of two old values, r
times the guy here.
So in other words, here's my little molecule.
This is the new value.
It comes from r times this plus what,
you have to tell me what multiplies
this u at this point.
So here's u_(j, n+1), and here's u_(j+1, n), and here is u_(j,
n).
And what goes in parentheses?
1 minus r.
Because I get the 1 when that flips over to the other side.
Good.
So it's just a combination.
The new value is a combination of those two.
And I guess what I want to show is that it could not possibly
be stable unless r is less or equal to 1.
So this will be the Courant condition.
The condition on the Courant number.
So the Courant condition or the CFL --
that's a really a better condition -- will be --
let me say what it is.
The CFL condition is r less or equal to 1.
That's a condition for stability or convergence
that comes to us from comparing the real characteristics
with the finite difference picture.
OK.
So what's CFL?
C is Courant still.
So this came from a really early paper,
and I think actually Lewy.
That's Courant, that's Friedrichs,
they were close friends.
That's Lewy, all three were friends actually.
And I think it was actually Lewy who spotted what we're just
going to do right now, the requirement that r
had to be less or equal 1.
Now let me just make clear that this Courant-Friedrichs-Lewy
condition is going to be a necessary condition.
It has to hold, or there is no hope.
But it's not enough, it's not a sufficient condition.
We can't guarantee that a method stable just because the Courant
condition is satisfied.
The Courant condition will tell us
something about the position of characteristics,
we'll see it in a second.
For example, this centered one is unstable, this guy,
when the right-hand side -- you might say,
oh this is a better method; always better to take
a centered difference for du/dx than a one-sided.
But not now.
The one-sided one is OK.
The centered one is going to be totally unstable,
for all r, this centered one.
And we'll identify that not from Courant-Friedrichs-Lewy --
their paper wouldn't have spotted the difference between
one and two, but von Neumann did;
von Neumann quickly noticed that if you use an exponential --
so we'll do one van Neumann in a moment --
if you plug in an exponential to number 2,
this centered difference, it takes off.
And it's unstable even if delta t is small.
OK.
Now what is this CFL, Courant-Friedrichs-Lewy
condition?
Where does that come from?
It comes from just thinking about the information.
I have to write down the thinking now
that goes into the Courant-Friedrichs-Lewy
condition, the CFL condition.
OK.
It's straightforward.
I have to think, at a time t, let's say t equals n delta t.
Suppose I've taken n times steps.
Then the true u at, let's say, 0 and that time t is --
or x and t, because we know the true solution.
The true u at x and t is, as we know,
the initial function at the point x plus c*t.
So that's the correct solution.
And Courant-Friedrichs-Lewy are just
saying that you better use this number in finite difference
method or you don't have a chance.
Right.
That makes sense.
So what do I mean by use this number?
OK.
So let me draw.
So u at x and t is some point here.
This is at the point x and t.
This used these two values.
They use these three values.
This one used that one, that one, that one.
This one used those two.
Those four used these five values.
Right.
See, the difference method is not propagating the information
entirely on that line, it's really taking values
over that whole interval with some combinations of r
and 1 minus r.
And in the end, after five times steps,
it's giving us this value up there.
But the point is that if this point x plus c*t were out here,
then refining delta x and refining delta t,
and using more and more points, and filling in and using more
of these, but not getting out here, would get you nowhere.
You wouldn't have a chance.
So that's the Courant-Friedrichs-Lewy
condition.
How far out is this?
This is n delta x inside this.
And if this is x, this is n delta x's away,
and this is x plus c*t.
And c is n delta t's.
Right?
t is n delta t.
Are you with me?
As I keep delta t over delta x, the ratio r, fixed
and use more and more points, I'm filling up this interval
with points that I've used.
But I'm never going to get to a point there.
So the idea is then, unstable, couldn't converge,
couldn't work, if the distance reached here, n delta x,
is smaller than c*n delta t, it couldn't work.
OK.
So I'm taking a negative point of view here.
It can't work in this case.
The reason I take that negative point of view is I'm
not able to say it does work in the case when
delta x is big enough compared to delta t,
and I include that point.
I can't be sure.
But I can be sure that if I don't include
that neighborhood of where the true u is produced,
it can't have a chance.
So let me just cancel n, divide by delta x.
And I say fail if 1 is smaller then c delta t
over delta x, which is r.
It fails if r is bigger than 1, by this reasoning
of characteristics.
There are two more cases that are worth thinking about.
Suppose the method was downwind.
So this is upwind, because I take a forward difference.
The wind is blowing this way.
Forgive me for incomplete use of the word wind.
I don't know what it means.
But one way or another, whatever, the wind
is blowing this way.
The values are coming from the upwind direction.
And that's good, because the true value comes
from the upwind direction.
That's what we saw in characteristics,
the values are blowing downwind.
So, what would happen if I use a backward difference here?
So I didn't write that bad idea, but it's
important to realizes that's method 1a, a bad idea,
is to use a backward difference here.
I should maybe call it 1b, for backward.
You see that it would fail?
If I used a backward difference, what will
what happen to this picture?
This value would come from stuff on the left.
It would totally have no chance to use
the correct initial value.
So a backward difference is an immediate failure.
So r has to be -- It will also fail if --
well I was going to say r less than 0.
I don't know, can I say that?
The way r could be less than 0 would be the delta x being
negative, and that's what I mean by the downwind method,
where I'm looking for looking for the value here,
and of course I'm not finding it.
OK.
So I guess I'm saying that Courant-Friedrichs-Lewy tells
us right away that this method will only have a chance --
it only has a chance if r is -- so, possible,
and in fact this is the stability condition.
So let me write that fact, but we haven't proved it yet.
It's stable for r between 0 and 1.
That's the stability condition, which limits delta t.
Because r is delta t over delta x.
And this says that delta t can't be larger
than the multiple of delta x.
OK.
You might say, that sounds like delta t is too small.
Is stability too restrictive?
Well it's restrictive certainly, because it limits delta t.
But actually, for these methods, accuracy also limits delta t.
If I look for an implicit method, that would allow me --
if I looked for a method that allowed a bigger delta t,
then the error of that, in that method,
proportional to delta t, would be too big.
In other words, accuracy as well as
stability is keeping delta t of the order of delta x.
Accuracy as well as stability is keeping r within a bound,
and stability is keeping it within those bounds.
OK.
So I still have to show that the method is stable.
The Courant-Friedrichs-Lewy condition
just helps to eliminate impossible ratios.
OK.
One more point, which of course everybody notices, in solving
the difference equation.
Suppose r is exactly 1.
Think about the case where r is exactly 1.
Remember that's c delta t over delta x.
Suppose r is exactly 1, then what does the method do?
The method takes the new value, r is 1.
This is 0 now.
So the new value is then, when r is 1, is that old value.
In this picture, when r is 1 this value is this one.
The values are traveling along a line.
I'm not using these you see, because if r is exactly 1,
these are exactly 0.
Now is that good?
Actually it's terrific, because that line, when r is 1,
is the correct characteristic line.
If r is 1, if c delta t equals delta x,
I'm going up exactly with that slope that the true line does.
In other words, the difference equation
gives exactly the solution to the differential equation.
Because it's so simple of course.
Couldn't do it -- well, that's the point,
we can't do it for big problems, because in a big problem c is
going to be variable.
Maybe the problem, c may depend on the position or the time,
or it might depend on the solution
u in a non-linear problem.
Those are ahead of us.
But this possibility in 1D -- it's only, really,
a one-dimensional possibility, where the true information
travels on a line, and we could just hope to stay close to that
characteristic line.
OK.
Now I'm ready to tackle, for any method, the --
let me get these guys up again.
So first, I want to see, why is upwind stable,
and why is centered unstable.
Should I start with the stability question, rather than
the accuracy question?
The accuracy we could guess was first order,
until we get to Lax-Wendroff.
So accuracy is going to come next time.
We've had Courant's take on stability, the CFL condition,
but now I'm ready for von Neumann's deeper insight.
OK.
So how does von Neumann study stability?
He watches every e to the i*k*x.
So von Neumann's test is going to be each e to the i*k*x
should have a growth factor, in the difference equation,
smaller then 1.
The growth factor in the differential equation,
of course, was right on.
This has magnitude exactly 1.
So wave equations are not giving us any space to work in.
Because we want -- the growth factor and the difference
equation, we want it to be close to the real one.
And at the same time, it better be not -- the real one,
having absolute value exactly 1 on the unit circle.
But the difference equation is allowed
to go into the unit circle, but not to go out.
OK.
That will separate the good ones from the bad.
OK.
Let me work here on -- Can I copy the equation u_(j,
n+1) is equal to r*u_(j+1, n), and 1 minus r u_(j, n).
OK.
Now I'm going to do an exponential again.
So I'm going to have u, say, 0, n.
I'm going to start from e to the i*k*x.
Sorry, let me say -- So, time 0 and position n will be e
to the i*k*n delta x.
Right.
That's my initial, e to the i*k*x,
that's my pure frequency.
I plug that in to the -- oh, not n but j;
j is measuring how many -- sorry, let's get it right --
j is measuring how many delta x's I'm going across;
n is measuring how many delta t's I'm going up.
So this is the key, here's von Neumann.
Try a pure exponential, plug it in.
Again, it's going to work, because we
have constant coefficients.
So what do I get on the right-hand side?
I get r times this guy at level j plus 1,
e to the i*k j plus 1 delta x times the exponential.
Can I save the exponential, because it's
going to factor out?
Plus a 1 minus 4 times the exponential itself,
and here's the exponential: e to the i*k*j delta x.
This will be one step.
I could say u_(j, 1) coming from level 0 to level 1.
Sorry, e to the i*k*j delta x, so I don't need it here.
Sorry, I did it wrong.
That e to the i*k j plus 1 delta x factors into e to the i*k
delta x from the 1, and e to the i*k*j delta x,
which is over there.
So I only wanted this much.
OK.
Sorry, that's on the permanent tape but maybe it's OK.
Because it makes us think through,
what's the result from a pure exponential?
And again, the result is pure exponential
in, pure exponential out, but multiplied
by some finite difference growth factor G, that
depends on the frequency.
And it depends on delta x and it depends on r.
Well actually it depends on k delta x together.
So I could put that together.
What's von Neumann's question now?
He wants to know is this number -- it's a complex number,
of course, because it's got this cosine of k delta x plus i sine
of k delta x -- he wants to know is it -- magnitude,
does the magnitude get bigger than 1?
If so, n steps will give the n-th power of the thing,
and it will blow up.
Or does the magnitude stay lesser
equal to 1, in which case it won't blow up, in which case
we have stability.
Of course, it would be great if the magnitude was always
exactly 1.
Because that's what the true solution has.
But that's only going to happen in this special, special case
when r equals 1, and I'm going just right
on the characteristic.
When r is exactly 1, that's the case when that's gone,
this is a 1, and that has magnitude exactly 1.
But normally, r is going to be, I'm
going to be on the safe side, r will be less than 1.
And I just want to draw this von Neumann amplification factor.
I'll often use the word growth factor,
just because it's shorter, but another word
is amplification factor.
What does an exponential get amplified by?
Can you identify this number in the complex plane?
Of course, the big question is, where is it
with respect to the unit circle?
OK.
Take k equals 0 -- zero frequency, the DC term,
the constant start -- what is that number equal when k is 0?
1.
So that's normal.
At k equals 0, we're right at the position 1.
That's just telling us that a constant initial value
stays unchanged.
The true growth factor and the von Neumann amplification
factor both 1.
But now let k be non-zero.
Where is this complex number?
And now, I'm going to impose the condition
that r is between 0 and 1, because I know from Courant
that is going to be required.
What's the point here?
OK.
Then, you see, here is a 1 minus r.
So there's 1, 1 minus r will be a little bit in here,
and then what happens when I add on this number?
So the 1 minus r I've done.
It put me here.
It was real.
But this number is not real.
This number is r times some complex number,
but what do I know?
I do something critical here.
What do I know about this number?
I know its absolute value is 1.
So that it gets multiplied by an r.
Look, look, look, look.
I start with a 1 minus r.
So I go to 1, back to r, and now I
add in this part, which is somewhere on a circle of radius
r that's everything in this problem.
It's a circle of radius r around the point 1 minus r.
I'm inside the circle, and you could say, well,
you could have seen that clearly by just using the triangle
inequality.
The magnitude of this can't be bigger than the magnitude
of that, which is what?
r.
Plus the magnitude of that, which is what?
1 minus r.
So the magnitude, by the triangle inequality,
can't be more than r plus 1 minus r which is 1.
But now wait a minute, where did I use the Courant condition?
The way I said that there sounded
like it would always work.
This has magnitude r.
What's going on here?
Suppose r is bigger than 1, what's going wrong?
In fact, what's the picture if r is bigger than 1?
If r is bigger than 1, then my 1 minus r is out here somewhere.
So this is the unstable case, 1 minus r,
and then a circle of radius r, bad news, right?
Every frequency is unstable in fact.
When I'm reaching too far, and that's just telling me
I'm not using the information I need.
I don't have a chance of keeping, controlling
any frequency.
OK.
I've got just enough time to show
how bad this second method is.
So nobody wanted his name associated
with that second method.
Why is that?
So what's the von Neumann quantity for the second method?
OK.
Just space here to write -- so this is now --
I want to know the number for that second method.
So now the method is u_(j, n+1) is u_(j, n),
from the time difference, plus r over 2, u_(j+1,
n) minus u_(j-1, n).
This is going to be the bad method.
It looks good because the centered difference
is more accurate than a one-sided difference,
but it's also unstable here.
Can you look at that and see what the von Neumann number
G is going to be?
Think of an exponential going in.
What exponential comes out?
So G is a 1 from this.
Now this is our guys that's shifted over,
so that would be in r over 2 e to the i*k delta x,
just as it was before.
And this one will be a minus r over 2 times e to the --
so it's back one step, so there will be an e to the minus i*k
delta x.
So what's with this now?
G is 1 plus this quantity, r over 2 times e to the i*k*x
minus e to the minus i*k*x.
What am I seeing there?
I'm seeing 1 plus r, and everybody recognizes this minus
this over 2 as i sine, i*r sine k delta x.
That's the amplification factor, and is it smaller than 1?
No way.
Let me raise it up here.
So I don't care whether r is less than 1 or not, I'm lost.
This is a pure imaginary number.
This is a real number.
So I have the sum of squares to get the magnitude,
it will be the square root of 1 plus r sine k delta x squared.
So if I draw the bad picture then, what's the bad picture?
Is now, it goes to 1 and then it goes way up the imaginary axis.
Well maybe not way up, but up.
Sorry, I shouldn't have made it quite as bad as it was.
If I reduce r, I don't go so far up, but no hope.
So do you see why that one is bad,
because the amplification factor --
von Neumann tells us what Courant did not tell us,
that the amplification factor here has magnitude bigger than
1.
It's outside the circle, and every exponential is growing
and there's no hope.
OK.
So the second last lecture on this section 5-2 of the notes
will be about Lax-Friedrichs and Lax-Wendroff,
order of accuracy, stability and actual behavior in practice.
OK.
See you Wednesday.
Thanks.