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Welcome back.

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We're gonna switch gear a little bit now

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and move away from sort
of mathematical treatment

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of this continuous model

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and think about how to
implement them computationally.

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So as I said in the previous video,

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one of the big strengths
of differential equations

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is that they give you, without
any additional alteration,

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a complete view of the flow diagram

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because that's literally

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what those differential equations are

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is the flow or the speed of the system

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at any point in variable space.

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So if you tell me the
state, x of a system,

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the time derivative of x
of t gives me the direction

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of the movement of the
system, as well as its speed,

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how fast is it going there?

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I have three different
notations right now.

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So you're probably most used

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to the explicit time derivative.

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As I said, I like using Newton's notation.

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It's just a little simpler,

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this dot representing a time derivative,

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and really the way we write
those differential equation

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is having x dot or s dot
or i dot, whatever it is,

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equal to some function of
the state of the system x,

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maybe also explicitly of time, t.

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So these differential
equations are always written

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as x dot equals f of x of t.

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That's our differential,

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our system of differential equations.

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So in this video, what
I'd like to describe is

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what you'll be doing often,

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which is not necessarily
mathematical analysis

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over these differential equations,

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but just tracking them in time.

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So you put the system
in some initial state,

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you have some initial
conditions, x at t equals zero,

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and you wanna follow
this flow of dynamics.

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So let's say that our system here

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has only one degree of freedom.

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This x of t is just one variable,

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and we're interested in knowing from x0.

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An initial condition of T equals zero,

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the state of the system
for all t, for all time.

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Now there might be some true
evolution for this system,

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which might look like this,

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but that would be the true x of t.

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The simplest way to follow it numerically

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is discretized time,
the same way we're doing

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in discretized model.

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So at t equal zero,

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you can evaluate your
differential equation, x dot of t.

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So you evaluate this
function, f of x of t here,

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and what it gives you is
the instantaneous derivative

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at that point.

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So then one thing you can
do once you have that is say

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x at time, t, plus h
where h is some time step,

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simply gonna be equal to where I am

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plus the projection over h of the slope,

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f of x of t, my differential equation.

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And then that might put me here.

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And then I reevaluate my equation,

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project myself, reevaluate
project, project, project.

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And as you can see,

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I don't expect to get exactly
the right distribution.

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I mean, I'm doing a little
bit of whatever here,

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but the point is I expect to get something

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that's close to the true
solution, but that deviates.

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So this method, the simplest method

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is called Euler's method.

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And we're gonna say that its local error

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is of order, it's a big
O here, order h squared.

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The reason for that is we know
that there's a true x of t,

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and then there's the x
of t from Euler's method.

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And we can write x of t plus h

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by developing around x of t,

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the Taylor series of the true solution.

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So we know that it's
gonna be x of t plus h

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f of x of t plus...

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While we're doing that step,

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we know that the system
continues evolving,

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continues following the dynamics.

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So there is some curvature,

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curvature f prime of x of t plus h cubed.

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If you remember your Taylor series,

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three factorials, f prime
prime of x of t, and so on.

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And so the reason why we
say that Euler's method

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as a local error term in h squared

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is because it captures those first term

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of the Taylor series,

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but ignores the higher order term,

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so ignored by Euler.

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And so at every time step h that we do,

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the biggest error term is
gonna be this one here.

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So if the curvature of f
prime is equal to zero,

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if we have a linear growth
of a system, let's say,

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so f prime is gonna be zero.

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Euler's method is gonna do awesome

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because this error term is equal to zero,

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but for any like non-zero error,

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really the key point is that

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the biggest term that we have
is proportional to h squared,

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and that means two things.

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That means just as we move
from discrete to continuous.

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If h goes to zero, if we
have a very good computer

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and we're very patient
and we say I'm gonna do,

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you know, infinitely many step

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of tiny, tiny, tiny, tiny durations,

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then Euler's method is gonna be fine,

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but otherwise, you know,
it's important to remember

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that you're going to have
those local error terms.

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And often, the difference
here, let me put it in blue,

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so the difference here is the local error.

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And often as in my cartoon,
this error sort of grows

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with time steps, right?

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It might shrink sometimes.

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You might be lucky,

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but overall you expect it to grow.

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And so, we're often gonna
say that the global error

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of our numerical integrator is the sum.

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Let me rewrite that more clearly here.

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Global error is gonna be the sum

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over all time steps of the local error.

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And we had the fact...

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So we had this idea that the local error

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was an h square for Euler.

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Well, this is gonna be equal

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to the duration of our
numerical integration.

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So we're integrating from t equals zero

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to, let's say T, capital T.

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So how many steps are we gonna have to do?

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How many local error are we solving here,

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are we summing up?

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So it's capital T divided by h.

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That's the number of
steps that we have to do,

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times local error, which
is of order h squared.

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So this is gonna be an h.

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And what I'm trying to say here is that

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there's this key thing is,

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which is that Euler's method
is capturing up to order h

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the Taylor series of the true solution,

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and it's so that its local error

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and what's being ignored is h square,

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but the global error,

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because you're doing a
lot of different steps,

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the global error is
still gonna be of order h

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even though you're capturing the thing.

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You're capturing the term of order h.

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Really, I wanna highlight this distinction

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between local error and global error

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because it is kind of
surprising to have something

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that captures the derivative.

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And yet as you integrate
a trajectory in time,

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you do expect its error to scale

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with your step size linearly.

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So basically if you wanna
do a numerical integration

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that's 10 times closer
to the true solution,

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if I wanted my red that's
here to be 10 times closer,

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I would need to do 10 times more steps.

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But we can have integrator that do better.

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So, I apologize here.

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I expected a blank sheet.

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So we wanna integrate
again from t equals zero

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to t equals T,

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whoops, x of t.

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We have something
different, true x of t here.

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And now we want to do better.

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Well, what's the key part that
Euler's method is missing?

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It's the fact that as you do one step,

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your derivative is already changing

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during that step, right?

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There might be a second derivative.

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And so it's a mistake
to to project it too far

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into the future.

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So there is another
method, which I've learned

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under the name of improved Euler.

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Improved Euler's method.

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If you Google around, in English,

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I believe it's often called Heun's method.

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Whoops.

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Too sensitive here.

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Heun's method.

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And this, I'll tell you right away

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is a second order method in global error.

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So this method is gonna
have a global error

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that scales of the order of h squared,

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meaning that, you know, if
you wanted Euler's method

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to be 100 times more accurate,

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you needed 100 times more time step.

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Here, you would only need
10 times more time steps

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because the global error
scales better with h.

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And so that means that the local error

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for the same reason as
before has to be an h cubed

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because the number of
steps we're making grows

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as one over h as we
decrease our time step.

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So the local error error
and the global error,

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there's one difference in order, okay.

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Well, what is this method?

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Let me just fix my typo here.

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Heun's method.

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The method itself looks super simple.

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What we're gonna say,
and I'm going to write it

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as xh for Heun at time t plus h,

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it's gonna be equal to x of t

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plus h f of x of t

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plus f of xe time t plus h.

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So I'm switching up my
notation here a little more.

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So let me explain what I mean,

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so that the prediction from Heun's method

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is gonna be a projection from
x of t over a time step h

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of the average between
the measured derivative.

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So this term here, this f of x of t

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is the initial derivative here.

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And now what I have, this
f of xe, let me go back,

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is gonna be the prediction
from Euler's method.

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So basically what you do here is you say

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I look at my initial derivative,

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I project myself over time h.

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Okay, project myself over time h,

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so I get an Euler prediction of here.

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Well, now at this point,

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my derivative here is much faster,

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much steeper than it
was at my initial point.

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So I'm gonna take the average

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of both, my first point
and my second point.

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So my average might look
something like this,

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and now this is what I'm gonna project,

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and I'm gonna end up here.

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So basically every step of Heun's method

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does one step with Euler's method,

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looks at the derivative and average.

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The derivative that Euler's would have

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for its second step with the initial step

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and then does a true
first step under Heun.

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Let me give another example here.

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So with Euler, I evaluate
my derivative here.

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I project into the future up to 2h,

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which would be, you know, somewhere here.

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Euler is gonna end up way off base here.

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It's a steep derivative,
might end up here.

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Now would evaluate the derivative
here, which is negative.

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So Heun would average both,

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which would be pretty
much flat in this case

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and end up here.

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Doing much better than
the blue dot, right?

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So Euler is doing this and this,

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and Heun is doing this and this.

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And the key part is that the local error

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and the global error are both
one order of magnitude better.

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So if you want to be
100 times more accurate,

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you only have to do 10 times more steps.

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Now every single evaluation
takes two evaluation

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of your derivatives.

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So every step does take
twice as many operation,

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twice as many evaluation
of this f function

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for your derivatives.

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But really you're still winning,
you know, a factor of five

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if you wanna be 100 times more accurate.

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So you can have your time
steps be 10 times smaller.

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You do twice as many operations.

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So that's 20 operations for
your 100 fold precision,

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which is still much better

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than what you would do with Euler.

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Now as you'll see in the homework,

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Heun's method, which is
this definition here,

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is actually not defined as
a simple arithmetic average

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between f of x of t and
f of xe of t plus h.

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The basic idea looks like
you're just taking an average,

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but that's not what it is.

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The goal here, and this is
a hint for the homework,

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is that you can ask, well, what is f of xe

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at time t plus h?

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You can develop that
with the Taylor series.

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And then you can show that this
one half is actually chosen

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such that this xh approximation

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at time t plus h captures one more term,

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so captures the h square term

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in the original Taylor
series for the true solution.

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This is something that like sounds weird.

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Like your only evaluating
first order derivative.

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How could you like go and get information

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about the second order derivative?

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But if you do work it out
by using Taylor series

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of both x of t, which we did right here,

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so this is our Taylor series
for the true solution,

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and then you go do your
Taylor series for this term,

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for the derivative itself
under Euler's method,

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then you can see that you're able

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to combine multiple
evaluation of your derivatives

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in a smart way to capture more and more

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or capture higher order terms.

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In a future video, so next week,

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I'll go into deeper detail
in the in the Runge-Kutta

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or Dormand-Prince method,

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which captures up to order four terms,

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and there it's just gonna
be a lot of arithmetics,

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but we're gonna be able
to see that similarly

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by doing four evaluations or derivative,

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we can get like a higher order term,

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and there it's gonna look
nothing like an average.

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So the fact that you have
a simple average here,

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this divided by two for both evaluation

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of the derivative is sort of coincidence,

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but it's a nice explanation.

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And as you can see, like in any cartoon,

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like the one in the top left here,

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it's clear that it's gonna do better.

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So, apologies, by the
way, if my picture here

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was, in the way for some
of the words up there.

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But, you know, this is a topic

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that we're gonna revisit
time and time again

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because the key part is
that even if you ask,

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you know, a scientific library,

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please integrate my differential
equations through time,

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it's super important to know what order,

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how much error are you getting,

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what's the order of the
method that it's using

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by default or that
you're asking it to solve

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because what comes out
of discrete computer

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and discrete integration
is not the true x of t,

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but an approximation of it.

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And as I showed in the cartoon,

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those approximations can be way off.

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So it it's gonna be important
to distinguish error

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due to our model specification

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and error due to how we
solve the model numerically.

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And one way to test how well we're doing

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is gonna be to compare our predictions

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from the mathematical models

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with simulations of the results.

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So in the next video,
we're gonna see how to use

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some of the tools we quickly saw

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to simulate discrete models,

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but this time in continuous time.

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So I'll see in the next one.