r-k method
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Next: Algorithm: Runge-Kutta Method of Up: Numerical Methods Previous: Error
Estimates and Convergence Contents
Runge-Kutta MethodRunge-Kutta Method is a more general and improvised method as compared to that of
the Euler's method. It uses, as we shall see, Taylor's expansion of a ``smooth function"
(thereby, we mean that the derivatives exist and are continuous up to certain desired
order). Before we proceed further, the following questions may arise in our mind,
which has not found place in our discussion so far.
1. How does one choose the starting values, sometimes called starters that are
required for implementing an algorithm?
2. Is it desirable to change the step size (or the length of the interval) during
the computation if the error estimates demands a change as a function of ?
For the present, the discussion about Question is not taken up. We try to look more
on Question in the ensuing discussion. There are many self-starter methods, like the
Euler method which uses the initial condition. But these methods are normally not
very efficient since the error bounds may not be ``good enough". We have seen in
Theorem that the local error (neglecting the rounding-off error) is in the
Euler's algorithm. This shows that as the values of become smaller, the
approximations improve. Moreover, the error of order , may not be sufficiently
accurate for many problems. So, we look into a few methods where the error is of
higher order. They are Runge-Kutta (in short R-K) methods. Let us analyze how the
algorithm is reacher before we actually state it. To do so, we consider the IVP
Define with and . We now
assume that and are smooth. Using Taylor's series, we now have
(14.3.10)
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For consider the expression
(14.3.11)
and are constants. When , ( ) reduces to the Euler's algorithm. We
choose and so that the local truncation error is . From the definition
of , we have
where denote the partial derivatives of with respect to
respectively. Substituting these values in ( ), we have
(14.3.12
)
A comparision of ( ) and ( ), leads to the choice of
(14.3.13)
in order that the powers of up to match (in some sense) in the approximate
values of . Here we note that . So, we choose and so that
( ) is satisfied. One of the simplest solution is
Thus we are lead to define
(14.3.14)
Evaluation of by ( ) is called the Runge-Kutta method of order (R-K method
of order ).
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A few things are worthwhile to be noted in the above discussion. Firstly, we need the
existence of partial derivatives of up to order for R-K method of order . For
higher order methods, we need to be more smooth. Secondly, we note that the local
truncation error (in R-K method of order ) is of order . Again, we remind the
readers that the round-off error in the case of implementation has not been considered.
Also, in ( ), the partial derivatives of do not appear. In short, we are likely to get a
better accuracy in Runge-Kutta method of order in comparision with the Euler's
method. Formally, we state the Runge-Kutta method of order .
Algorithm: Runge-Kutta Method of Order
For the IVP ( ) or ( ), let where is a given prescribed
step size. For , define by
Then the flow chart associated with the R-K method of order is
Figure: Flow-
Chart of Runge-Kutta method of order
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EXAMPLE 14.3.1 Use the Ringe-Kutta method to find the approximate value
of where is the solution of the IVP
with step sizes and Also, calculate the error and tabulate the
results.
Solution: Comparing the given IVP with ( ), we note that
and . We now calculate the values of and from the R-K method of
order and use the formula to calculate the approximate values.
The results are shown in Tables and .
Table
InitialInitial
Step sizeApprox Exact
Error
1.00000 0.10000 1.00000 1.00000 0.00000 0.1 0.121
1.00000 0.10000 1.11050 1.11111 0.1 0.146531
0.20000 1.11050 0.10000 1.23377 1.25000 0.01623 0.12332 0.18417
0.30000 1.23377 0.10000 1.338751 1.42857 0.04106 0.15222 0.23708
0.40000 1.38751 0.10000 1.58216 1.66667 0.08451 0.19252 0.31495
0.50000 1.58216 0.10000 1.83589 2.00000 0.16411 0.25032 0.00627
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Table
InitialInitial
Step sizeApprox Exact
Error
1.00000 0.05000 1.00000 1.00000 0.00000 0.05000 0.05513
1.00000 0.05000 1.05256 1.05263 0.05539 0.06138
0.10000 1.05256 0.05000 1.11095 1.11111 0.00016
0.06171 0.06876
0.15000 1.11095 0.05000 1.17618 1.17647 0.00029 0.06917 0.07755
0.20000 1.17618 0.05000 1.24954 1.25000 0.00046 0.0781 0.08813
0.25000 1.24954 0.05000 1.33264 1.33333 0.00070 0.08880 0.10102
0.30000 1.33264 0.05000 1.42755 1.42857 0.00102
0.10190 0.11696
0.35000 1.42755 0.05000 1.53697 1.53846 0.00149
0.11812 0.13697
0.40000 1.53697 0.05000 1.66451 1.66667 0.00215
0.13853 0.16255
0.45000 1.66451 0.05000 1.81505 1.81818 0.00313
0.16472 0.19598
0.50000 1.81505 0.05000 1.99540 2.00000 0.00460
0.199082 0.24079
Runge-Kutta Method of Order
We are now ready to state the algorithm for using the Runge-Kutta method of
order . This is a generalization of the R-K method of order to higher order
methods. Without getting into analytical details, we state the R-K method of order
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. It is widely used algorithm. For the IVPs ( ), set for
where . Also, we set . The for , we define
where
Remark 14.3.2 The local error in the R-K method of order is . To achieve
this error, we are forced to do more computation or in other words, spend more time
to compute and . It all depends on the nature of the function to estimate
the time consumed for the computation. The cost we pay for higher accuracy is more
computation. Also, to reduce the local error, we need smaller values of the step
size , which again results in large number of computation. Each computation leads
to more of rounding errors. In other words, reduction in discretisation error may lead
to increase in rounding off error. T HE MORAL IS THAT THE INDISCRIMINATE REDUCTION OF STEP-SIZE NEED NOT MEAN MORE ACCURACY .
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Figure: Flow-Chart of Runge-Kutta method of order
EXERCISE 14.3.3 Use the Runge-Kutta method of order to find an approximatesolution of the IVP
with step size . Also, calculate the error and tabulate the results.