Runge kutta method

What is Numerical Analysis?

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis.

Definition of Numerical Methods:

A numerical method is a complete and unambiguous set of procedures for the solution of a problem, together with computable error estimates

Numerical methods are algorithms used for computing numeric data. They are used to provide ‘approximate’ results for the problems being dealt with and their necessity is felt when it becomes impossible or extremely difficult to solve a given problem analytically.

It is important to recognize under what conditions a method can be followed and what starting value(s) to choose from in order to ensure that the chosen method shall work(converge).

Numerical methods can be used for-

• finding root(s) of equations – Bisection method, Newton Raphson, Fixed Point iteration etc

• solving ODEs – Euler method, Improved Euler, RK methods, Mid point method, Predictor Corrector methods etc

The one which is so accurate, that most computer packages designed to find numerical solutions for differential equations will use it by default—the fourth order Runge-Kutta Method.

What is meant by Runge-Kutta 4th Order Method?

A method of numerically integrating ordinary differential equations by using a trial step at the midpoint of an interval to cancel out lower-order error terms. The second-order formula is

(1)

(2)

(3)

For

Runge Kutta 4th order method is given by

where

• K1 is the increment based on the slope at the beginning of the interval, using y and K1 (Euler’s method);

• K2 is the increment based on the slope at the midpoint of the interval, using y and K1;

• K3 is again the increment based on the slope at the midpoint, but now using y and K2;

• K4 is the increment based on the slope at the end of the interval, using y and K3.

How to write Ordinary Differential Equation

How does one write a first order differential equation in the form of

Example-1:

is rewritten as

In this case

Example-2:

A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by

To find the temperature at seconds using Runge-Kutta 4th order method.

Assume a step size of seconds.

Solution:

Step 1:

is the approximate temperature at

q2 is the approximate temperature at

The exact solution of the ordinary differential equation is given by the solution of a non-linear equation as

The solution to this nonlinear equation at t=480 seconds is

Figure 1. Comparison of Runge-Kutta 4th order method with exact solution

Effect of step size

Table 1. Temperature at 480 seconds as a function of step size,

Step size,h q (480) Et |?t|%

480

240

120

60

30 ?90.278

594.91

646.16

647.54

647.57 737.85

52.660

1.4122

0.033626

0.00086900 113.94

8.1319

0.21807

0.0051926

0.00013419

(exact)

Effects of step size on Runge-Kutta 4th Order Method

Figure 2. Effect of step size in Runge-Kutta 4th order method

Runge – Kutta method is also implemented in medical field to predict the spread and impact of various diseases. Here we can compare the calculated data with the real data and conclude about the seriousness of disease.