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In this segment, we're going to talk
about the formulas for the Runge-Kutta second-order method.
So these are three different methods which are used
under the Runge-Kutta second-order method to solve ordinary differential equations
of the first-order.|So we're going to talk about just the formulas in
this segment here.|So again, let me refresh the memory that we are trying
to solve the differential equation which is of this form.
So it is a first-order ordinary differential equation of this particular form,
and the general formula for the Runge-Kutta second-order method is given as this, that
you want to calculate the value of y at a step ahead, will be given by
the value of y at the point where you are, plus a1
times k1, plus a2 times k2, times h,
where k1 is equal to the value of the function at
the point where you are, and k2 is the value of the function at some point
ahead, which is given by xi plus
p1 times h, yi plus q11
k1 times h.|So the beauty of
using the Runge-Kutta second-order method as follows is that we don't have to find
out the derivative of this function f anymore.|So we can still get the
second-order accuracy is our Runge-Kutta second-order method by simply calculating the value of the function
at these points.|However, the question arises that we do need to know what a1 is,
what a2 is, what p1 is, and what q11 is, because if we don't know
what those values are, then we won't be able to use this formula properly.|Now, what
Runge did was they took this particular expression and equated it to
the second-order . . . the first three terms of the Taylor series expansion
for y, and that's in a separate segment, and what they found out
is that they got three equations, they got a1 plus a2 equal to 1,
a2 p1 is equal to 1/2,
and a2 q11 is equal to 1/2, so that's what
they obtained, they obtained three equations, but we have four unknowns, keep in mind we have a1, a2, p1, and
q11, which are the four unknowns, but if you are able to meet the conditions
of these three equations, then you have achieved your second-order accuracy as per Runge-Kutta
second-order method.|So that's the reason why we have several
different methods of Runge-Kutta second-order method, because you have three equations and four unknowns, that
means that one of the choices can be made by you.|So there are some three
popular choices which people use, and the first popular
choice comes under the Heun's method.|In the Heun's method, what you are doing
is you are choosing a2 equal to 1/2.|So if you choose
a2 equal to 1/2, then a1 plus a2 equal to 1
will give you a1 is equal to 1/2.|The second equation,
a2 p1 is equal to 1,
will give you . . . 1/2, will give you p1
is equal to . . . it will give you p1 is equal to 1.
And then the last equation, which is
a2 q11 equal to 1/2, and that will give you
q11 is equal to 1 also.|So if you are just looking at that,
then the equations which you are writing for the Heun's method, which is part of the Runge-Kutta second-order
method, will look like this, since a1 is 1/2,
k1, plus 1/2 k2, times h, because a1 is 1/2,
and a2 is 1/2, and then your values for k1
will turn out to be the value of the function at xi, comma, yi,
and what is the value of k2 now?|It will be xi plus p1 times h,
because since p1 is 1, so it will be just h, yi
plus q11, q11 is 1, that's 1, plus
k1 times h.|So that's
what you get as the Heun's method, so that's what you are doing in the Heun's method.|Now, you
can very well see that, from a physical point of view, what you are basically doing in the Heun's method is that you are
calculating the value of the slope at the point where you are, and then you are calculating the value of the slope
where you want to go, which is xi plus h, which is the same as xi-plus-1, and then what
you are doing in the Heun's method is that you're taking the average of the two slopes, because you're giving half
the weightage to the slope at this point, and half the weightage for the slope at this particular point,
and then you are multiplying by h.|So that's the difference between Heun's method and your Euler's method,
that you are giving some value to the slope at xi,
and the slope at xi plus h, you give equal weightage to it, half and half,
and that's what you are using the calculate the new value of y.|Let's look at the other methods
which are there.|So you've got midpoint method.
.
In the midpoint method, a different value of a2 is chosen, a2 is chosen as 1.
So if a2 is chosen as 1, we go back to our two equations which we had, we had a1 plus a2
equal to 1, that will simply give you a1 is equal to 0.
The next equation is a2 p1 is equal to 1,
and that'll give you p1 is equal to 1.
And then . . . sorry, 1/2, so that'll give p1 is equal to 1/2.|And then we
have a2 q11 equal to 1/2, and since a2 is equal to 1,
q11 will be also 1/2.
So if we substitute these values back into the general formula for the Runge-Kutta second-order method, this is
what we're going to get, we're going to get y-sub-i-plus-1 is equal to yi,
plus, since a1 is 0 and a2 is 1,
I'm going to simply get k2 times h, and where, again, k1 is nothing but
the value of the function at the point where you are, and k2 is the value
of the function now at xi plus p1, p1 is 1/2
times h, and yi plus
q11 is 1/2, times k1 times h.
So again, what you are finding out here is that it is some way of finding the
slope of the function, of the dependent variable, which is being used in the
midpoint method.|Now, you are using k2, and k2 is
simply the value of the slope, some approximate value of the slope, at
the point which is somewhere in between xi and x-sub-i-plus-1, because it's exactly
1/2 h away from the point where you are, so it is, that's why it's called the midpoint rule, because
the slope which you are calculating is at the middle of xi and x-sub-i-plus-1.
So you want an xi here, this is x-sub-i-plus-1, and you're calculating the slope
right in the middle right there, so that's why it's called the midpoint method.|Now, some people might say that, hey,
if you look at this particular formula here, it has only k2 in it, k1 is not there, so why do I have to
calculate k1?|The reason why you have to calculate k1 is because it is used here to be able
to calculate the corresponding approximate value of y at this midpoint between
xi and x-sub-i-plus-1.|So let's look at the last . . .
last popular method which is used, which is called the Ralston's method.
In the Ralston's method,
what you are doing is that you're choosing a2 equal to 2/3.|So when you choose
a2 equal to 2/3, from your a1 plus a2 is equal to 1,
you'll get a1
equal to 1/3, and a2 p1
is equal to 1/2, and from there you'll p1 is equal to
3/4, and a2 q11 is also
1/2, and since a2 is 2/3, you'll get q11 also
equal to 3/4.|So from here, if I substitute it
back into the general Runge-Kutta second-order method, I'm going to get y-sub-i-plus-1
is equal to yi, plus a1, which is 1/3, k1,
plus a2, which is 2/3, k2, times h,
where k1 is equal to the value of the function at xi,
comma, yi, and k2 is the value of the function at xi,
plus
p1 times h, p1 is 3/4, 3 by 4 h, yi
plus q11, which is 3/4
k1 times h.|So again, the same
argument goes here, that what you are doing is that you are giving 1/3 of the weightage
of the slope to the . . . to the value of f at the point where you are,
and you're giving 2/3 weightage to the value of this slope, or the value of the slope
of y, or f at 3/4 away from the point where you are.
So again, all these methods which you are talking about, they're all based on giving some weightage to the slopes
at different points, but you have to calculate what those slopes are, and that's why you
need some approximate values of y to calculate this slope.|So here, for example, you are trying to calculate the value
of the function, or the slope of y with respect to x at this point, so you do need an approximate
value of y, which is given by this particular quantity here, and then you are giving 1/3 weightage
to k1 and 2/3 weightage to k2, and that's how you are able to calculate the value of the function at
some point ahead.|So these are the three most popular formulas for the Runge-Kutta second-order method, and
in the next segment, we'll use them to do an example.|And that's the end of this segment.
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