neub mat 201 lecture 12: initial value problems

NEUB MAT 201 Lecture 12: Initial Value Problems

Prepared BY Shahadat Hussain Parvez

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Initial value problems

Methods described here are for solving differential equations of the form:

𝑑𝑦

𝑑𝑥= 𝑓(𝑥, 𝑦)

The methods in this lecture are all one-step methods and have the general format:

𝑦𝑖+1 = 𝑦𝑖 + 𝜙ℎ

where ϕ is called an increment function, and is used to extrapolate from an old value yi to a new

value yi+1.

This is in a general function of

𝑁𝑒𝑤 𝑣𝑎𝑙𝑢𝑒 = 𝑂𝑙𝑑 𝑣𝑎𝑙𝑢𝑒 + 𝑠𝑙𝑜𝑝𝑒 × 𝑠𝑡𝑒𝑝 𝑠𝑖𝑧𝑒

According to this equation, the slope estimate of φ is used to extrapolate from an old value yi to a

new value yi+1 over a distance h (Fig. 1). This formula can be applied step by step to compute out into

the future and, hence, trace out the trajectory of the solution.

Figure 1 Graphical depiction of a one-step method.

All one-step methods can be expressed in the general form, with the only difference being the

manner in which the slope is estimated. As in the falling parachutist problem, the simplest approach

is to use the differential equation to estimate the slope in the form of the first derivative at xi. In

other words, the slope at the beginning of the interval is taken as an approximation of the average

slope over the whole interval. This approach, called Euler’s method, is discussed in the next section.

This is followed by other one-step methods that employ alternative slope estimates that result in

more accurate predictions. All these techniques are generally called Runge-Kutta (RK) methods.

Euler’s Method

The first derivative provides a direct estimate of the slope at xi:

ii

x

yxfdx

dy

i

,

and the Euler method uses that estimate as the increment function:

𝜙 = 𝑓(𝑥𝑖 , 𝑦𝑖)

where f (xi, yi) is the differential equation evaluated at xi and yi. This estimate can be substituted to

get

𝑦𝑖+1 = 𝑦𝑖 + 𝑓 𝑥𝑖 , 𝑦𝑖 ℎ



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This formula is referred to as Euler’s (or the Euler-Cauchy or the point-slope) method. A new value of

y is predicted using the slope (equal to the first derivative at the original value of x) to extrapolate

linearly over the step size h (Fig. 2).

Figure 2 Euler's method

Error analysis of Euler’s method

The numerical solution of ODEs involves two types of error:

– Truncation errors, caused by the nature of the techniques employed

– Roundoff errors, caused by the limited numbers of significant digits that can be retained

The total, or global truncation error can be further split into:

– local truncation error that results from an application method in question over a single

step, and

– propagated truncation error that results from the approximations produced during

previous steps.

The local truncation error for Euler’s method is O(h2) and proportional to the derivative of f(t,y)

while the global truncation error is O(h).

This means:

– The global error can be reduced by decreasing the step size, and

– Euler’s method will provide error-free predictions if the underlying function is linear.

Euler’s method is conditionally stable, depending on the size of h.

Higher order Taylor series method

One way to reduce the error of Euler’s method would be to include higher-order terms of the Taylor

series expansion in the solution. For example, including the second-order term

With a local truncation error of

Although the incorporation of higher-order terms is simple enough to implement for polynomials,

their inclusion is not so trivial when the ODE is more complicated. In particular, ODEs that are a

function of both the dependent and independent variable require chain-rule differentiation. For

example, the first derivative of f (x, y) is

The second derivative is

Higher-order derivatives become increasingly more complicated.



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Heun’s Method

One method to improve Euler’s method is to determine derivatives at the beginning and predicted

ending of the interval and average them:

Figure 3Graphical depiction of Heun’s method. (a) Predictor and (b) corrector.

This process relies on making a prediction of the new value of y, then correcting it based on the

slope calculated at that new value.

From Eulers method

𝑦𝑖′ = 𝑓(𝑥𝑖 , 𝑦𝑖)

Is used to extrapolate linearly to 𝑦𝑖+1:

𝑦𝑖+10 = 𝑦𝑖 + 𝑓 𝑥𝑖 , 𝑦𝑖 ℎ

For the standard Euler method we would stop at this point. However, in Heun’s method the y0i +1

calculated in last equation is not the final answer, but an intermediate prediction. This is why we

have distinguished it with a superscript 0. This equation is called a predictor equation.It provides an

estimate of yi+1 that allows the calculation of an estimated slope atthe end of the interval:

𝑦𝑖+1′ = 𝑓(𝑥𝑖+1 ,𝑦𝑖+1

0 )

Thus, the two slopes can be combined to obtain an average slope for the interval:

This average slope is then used to extrapolate linearly from yi to yi+l using Euler’s method:

This is called corrector equation

This predictor-corrector approach can be iterated to convergence:

Error can be calculated using

where 𝑦𝑖+1𝑗−1

𝑎𝑛𝑑 𝑦𝑖+1𝑗

are the result from the prior and the present iteration of the corrector,

respectively.



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Midpoint Method

Another improvement to Euler’s method is similar toHeun’s method, but predicts the slope at the

midpoint of an interval rather than at the end:

This method has a local truncation error of O(h3) and global error of O(h2)

Runge-Kutta (RK) Methods

Runge-Kutta (RK) methods achieve the accuracy of a Taylor series approach without requiring the

calculation of higher derivatives.

𝑦𝑖+1 = 𝑦𝑖 + 𝜙 𝑥𝑖 , 𝑦𝑖 , ℎ ℎ

For RK methods, the increment function ϕ can be generally written as:

nnkakaka 2211

where the a’s are constants and the k’s are

hkqhkqhkqyhpxfk

hkqhkqyhpxfk

hkqyhpxfk

yxfk

nnnnninin

ii

ii

ii

11,122,111,11

22212123

11112

1

,

,

,

,

where the p’s and q’s are constants.

Various types of Runge-Kutta methods can be devised by employing different numbers of terms in

the increment function as specified by n. Note that the first-order RK method with n = 1 is, in fact,

Euler’s method. Once n is chosen, values for the a’s, p’s,and q’s are evaluated

Classical Fourth-Order Runge-Kutta Method

The most popular RK methods are fourth-order, and the most commonly used form is:

hkkkkyy ii 43211 226

1

where:

hkyhxfk

hkyhxfk

hkyhxfk

yxfk

ii

ii

ii

ii

34

23

12

1

,2

1,

2

1

2

1,

2

1

,



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Figure 4Graphical depiction of the slope estimates comprising the fourth-order RK method.

Systems of Equations

Many practical problems require the solution of a system of equations:

nn

n

n

n

yyyxfdt

dy

yyyxfdt

dy

yyyxfdt

dy

,,,,

,,,,

,,,,

21

212

2

211

1

The solution of such a system requires that n initial conditions be known at the starting value of x.

Solution Methods

Single-equation methods can be used to solve systems of ODE’s as well; for example, Euler’s method

can be used on systems of equations - the one-step method is applied for every equation at each

step before proceeding to the next step.

Fourth-order Runge-Kutta methods can also be used, but care must be taken in calculating the k’s.

1. Chapra examples 25.1, 25.5, 25.7 [To be done in class]

2. Chapra examples 25.2-25.3, 25.6 [ Home task]

3. Chapra examples 25.4 [For Lab]

4. Chapra exercise 25.1- 25.5

Self study

25.1.2 Algorithm for Euler’s Method

25.2.3 Computer Algorithms for Heun and Midpoint Methods

25.3.5 Computer Algorithms for Runge-Kutta Methods

25.3.1 Second-Order Runge-Kutta Methods

25.3.2Third-Order Runge-Kutta Methods

25.3.4 Higher-Order Runge-Kutta Methods

25.4 SYSTEMS OF EQUATIONS

Reading Materials 1. Numerical Methods for Engineers by Steven C. Chapra and Raymond P. Canale, McGraw Hills

Education, 6th Edition

a. Chapter 25



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Examples to be done in class. Example 25.1:Euler’s Method



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Example 25.5:Heun’s Method



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Example 25.7



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The graph below shows

Figure 5 Comparison of percent relative error versus computational effort for first- through fifth-order RK methods

neub mat 201 lecture 12: initial value problems

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