Lecture 39 Numerical Analysis

Chapter 7 Ordinary Differential Equations

Introduction Taylor Series Euler Method Runge-Kutta Method Predictor Corrector Method

PREDICTOR – CORRECTOR METHOD

The methods presented so far are called single-step methods, where we have seen that the computation of y at t n+1 that is y n+1 requires the knowledge of y n only.

In predictor-corrector methods which we will discuss now, is also known as multi-step methods. To compute the value of y at t n+1, we must know the solution y at t n, t n-1, t n-2, etc.

Thus, a predictor formula is used to predict the value of y at t n+1 and then a corrector formula is used to improve the value of y n+1. Let us consider an IVP

Using simple Euler’s and modified Euler’s method, we can write down a simple predictor-corrector pair (P – C) as

Here, y n+1 (1) is the first corrected value of y n+1. The corrector formula may be used iteratively as defined below: The iteration is terminated when two successive iterates agree to the desired accuracy

In this pair, to extrapolate the value of y n+1, we have approximated the solution curve in the interval (t n, t n+1 ) by a straight line passing through (t n, y n ) and (t n+1, y n+1 ).

The accuracy of the predictor formula can be improved by considering a quadratic curve through the equally spaced points (t n-1, y n-1 ), (t n, y n ), (t n+1, y n+1 )

Suppose we fit a quadratic curve of the form where a, b, c are constants to be determined

As the curve passes through (t n-1, y n-1 ) and (t n, y n ) and satisfies we obtain Therefore

and Which gives or

Substituting these values of a, b and c into the quadratic equation, we get That is,

Thus, instead of considering the P-C pair, we may consider the P-C pair given by The essential difference between them is, the one given above is more accurate

However, this one can not be used to predict y n+1 for a given IVP, because its use require the knowledge of past two points. In such a situation, a R-K method is generally used to start the predictor method.

Milne’s Method It is also a multi-step method where we assume that the solution to the given IVP is known at the past four equally spaced point t 0, t 1, t 2 and t 3.

To derive Milne’s predictor- corrector pair, let us consider a typical IVP

On integration between the limits t 0 and t 4, we get

But we know from Newton’s forward difference formula where

Now, by changing the variable of integration (from t to s), the limits of integration also changes (from 0 to 4), and thus the above expression becomes

which simplifies to Substituting the differences It can be further simplified to

Alternatively, it can also be written as This is known as Milne’s predictor formula.

Similarly, integrating the original over the interval t 0 to t 2 or s = 0 to 2 and repeating the above steps, we get which is known as Milne’s corrector formula.

In general, Milne’s predictor- corrector pair can be written as

From these equations, we observe that the magnitude of the truncation error in corrector formula is while the truncation error in predictor formula is Thus: TE in, c-formula is less than the TE in p-formula.

In order to apply this P – C method to solve numerically any initial value problem, we first predict the value of y n+1 by means of predictor formula, where derivatives are computed using the given differential equation itself.

Using the predicted value y n+1, we calculate the derivativ y’ n+1 rom the given differential equation and then we use the corrector formula of the pair to have the corrected value of y n+1 Using the predicted value y n+1, we calculate the derivative y’ n+1 from the given differential equation and then we use the corrector formula of the pair to have the corrected value of y n+1

This in turn may be used to obtain improved value of y n+1 by using corrector again. This in turn may be used to obtain improved value of y n+1 by using the corrector again. This cycle is repeated until we achieve the required accuracy.

Example Find y (2.0) if y ( t ) is the solution of y (0) = 2, y (0.5) = 2.636, y (1.0) = and y(1.5) = Use Milne’s P-C method.

Solution Taking t 0 = 0.0, t 1 = 0.5, t 2 = 1.0, t 3 = 1.5 y 0, y 1, y 2 and y 3, are given, we have to compute y 4, the solution of the given differential equation corresponding to t =2.0

The Milne’s P – C pair is given as

From the given differential equation, We have, We have,

Now, using predictor formula, we compute

Using this predicted value, we shall compute the improved value of y 4 from corrector formula

Using the available predicted value y 4 and the initial values, we compute

Thus, the first corrected value of y 4 is given by

Suppose, we apply the corrector formula again, then we have Finally, y (2.0) = y 4 =

Example Tabulate the solution of in the interval [0, 0.4] with h = 0.1, using Milne’s P-C method.

Solution Milne’s P-C method demand the solution at first four points t 0, t 1, t 2 and t 3. As it is not a self – starting method, we shall use R-K method of fourth order to get the required solution and then switch over to Milne’s P – C method.

Thus, taking t 0 = 0, t 1 = 0.1, t 2 = 0.2, t 3 = 0.3 we get the corresponding y values using R–K method of 4 th order; that is y 0 = 1, y 1 = , y 2 = and y 3 = (Reference Lecture 38)

Now, we compute

Using Milne’s predictor formula

Before using corrector formula, we compute

Finally, using Milne’s corrector formula, we compute

The required solution is: t y

Lecture 39 Numerical Analysis