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This lecture is on solving equations on the
computer.
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So far we used graphical methods and analytical
methods to analyse x dot=f(x).
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We now consider some numerical integration
methods of x dot = f(x).
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Here is a general problem statement given
x dot = f(x) subject to the initial conditions
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x = x0 at t = t0, find a way to approximate
the solution x of t.
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Let us outline Euler’s methods: Initially
we are at x0 and the local velocity is f(x)
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not.
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If the phase point moves for a short time
delta t.
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The new position is x(t0) plus delta t is
approximately is equal to x0 plus f(x0) times
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delta t, which we call x.
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Then x(t0) plus delta t is approximately x1which
is equal to x0 plus f(x0) times delta t.
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So, the new location is actually x1.
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And so, the new velocity is f(x1) and so x2
= x1+ f(x1) times delta t and so we continue
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like this.
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So, the general update rule is then x of n
+ 1 =xn + f(xn) times delta t.
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So let see how Euler’s scheme actually works
plot x of t versus t we identify t0, t1 and
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t2 that is the exact solution we identified
x0 which is the initial conditions corresponding
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to t0 highlight x(t1) and x(t2).
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The open dot show values x(tn) at discreet
times tn = t0 + n times delta t.
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We then highlight values from Euler’s scheme
x1 and x2 that comes from Euler’s scheme
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and we go ahead and connect the dots.
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The close dots show the approximate values
given by Euler’s method note that delta
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t should be small else the approximation will
actually be bad.
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So here is a simple-minded representation
of Euler’s numerical schemes for approximating
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the solutions of a differential equations.
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So it is natural to ask, we can actually improve
on the Euler method.
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The issue with the Euler method is that it
estimates the derivatives only at the left-hand
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end of the time interval between tn and tn
+1.
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A better way is to use the average derivative
across the time interval.
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So here is the improved Euler method: using
the Euler method take a trial step across
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the interval and we will get a trial value
x tilde n +1 = xn + f(x)n times delta t.
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Then the average f(x)n and f(x) tilde n+1
and use it to make the real step across the
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interval.
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The method is as follows x tilde n+1 = xn
+ f(x)n times delta t, which is the trial
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step when xn+1 = xn + 1/2 of f(x)n + f(x)
tilde n+1 times delta t and this is the actual
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step.
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This gives the smaller error e which is x(tn)
- xn for a given step size delta t.
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Now in both the cases the error e tends to
0 as delta t tends to 0, but the error decreases
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faster for the improved Euler scheme.
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Note that the error is proportional to delta
t in the Euler method and it is proportional
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to delta t squared in the improved Euler scheme.
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Now other numerical methods that one could
use, now before mentioning them, we mention
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that in the language of numerical analysis
the Euler method is a first order method,
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and the improved Euler method is a second
order method.
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Higher order methods have been devised in
the literature, but they actually involve
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additional computations.
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In practice a very good scheme is the fourth
order Runge Kutta scheme.
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This was actually developed by German mathematicians
working in approximately 1900.
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So the objective is to find the xn+1 in terms
of xn, so xn+1 = xn+ one upon six times k1
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+ 2 times k2 + 2 times k3 + k4, where k1 is
f(xn) time delta t, k2 is f(x)n +1/2 k1 times
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delta t, k3= f(xn) + 1/2 k2 times delta t
and k4 = f(xn) + k3 times delta t.
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Now usually this gives us very accurate results
without having to rely on very small step
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sizes delta t.
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This lecture was centred around using the
computer to solve our differential equations
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using numerical methods.
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We seen variety of analytical method to develop
intuition about nonlinear equations of the
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form x dot = f(x).
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But it can sometimes rather difficult to develop
intuition purely analytically because non-linearity
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may be just very, very strange.
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So, it is perfectly fare and it is perfectly
sensible to actually use the computer to actually
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simulate the differential equation to actually
develop some insights about how the equation
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would actually behave.
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So to that end, we highlighted couple of numerical
schemes.
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We started off with very simple Euler methods,
we introduced you to the improved Euler method
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and we also mentioned that in practice a good
compromise between accuracy and efficiency
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is actually obtained by the fourth order Runge
Kutta method.
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Now interestingly these two mathematicians,
German mathematicians actually devised the
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scheme in 1901 that was way before computers,
were actually devised.
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In fact, computers played a extremely crucial
role in the popularisation of nonlinear dynamics
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and that was done by very famous paper by
Murray 1963, where he had numerical computations
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of a model, that he has devised for atmospheric
dynamics.
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To show that it had all kind of very strange
behaviour.
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So, the lesson from this particular lecture
is that numerical schemes and computer cannot
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be a substitute for an analysis.
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But in fact, there are excellent complements
to the analysis.