Question

mechanical engineering question

chapter 21 of advanced engineering mathematics 10th edition

Question: Discuss error at two of the methods taught in this chapter. Why does the error behave differently and what are the computational costs of reducing error?

You can use error at Euler's method and Improved Euler's method to answer the question. thank you

Answer 1

Experience with Euler’s1 method raises some interesting questions about numerical approxima-
tions for the solutions of differential equations.
1. What determines the amount of numerical error in an approximation?
2. Why is that halving the step size tends to decrease the numerical error in Euler’s method by one-half and the numerical error in the modified Euler method by one-quarter?
3. Are some differential equations more difficult to approximate numerically than others? If so, can this be predicted without doing numerical experiments?
None of these questions has a simple answer, but we will at least be able to offer a partial answer for each.

Any approximation of a function necessarily allows a possibility of deviation from the correct value of the function. Error is the term used to denote the amount by which an approximation
fails to equal the exact solution (exact solution minus approximation).2 Error occurs in an approximation for several reasons.

Truncation error in a numerical method is error that is caused by using simple approximations to represent exact mathematical formulas. The only way to completely avoid truncation
error is to use exact calculations. However, truncation error can be reduced by applying the same approximation to a larger number of smaller intervals or by switching to a better approx-
imation. Analysis of truncation error is the single most important source of information about the theoretical characteristics that distinguish better methods from poorer ones. With a com-
bination of theoretical analysis and numerical experiments, it is possible to estimate truncation error accurately.
Round-off error in a numerical method is error that is caused by using a discrete number of significant digits to represent real numbers on a computer. Since computers can retain a large
number of digits in a computation, round-off error is problematic only when the approximation requires that the computer subtract two numbers that are nearly identical. This is exactly
what happens if we apply an approximation to intervals that are too small. Thus, the effort to decrease truncation error can have the unintended consequence of introducing significant
round-off error.
Practitioners of numerical approximation are most concerned with truncation error, but they also try to restrict their efforts at decreasing truncation error to improvements that do
not introduce significant round-off error. In our study of approximation error, we consider only truncation error. We seek information about error on both a local and global scale. Local
truncation error is the amount of truncation error that occurs in one step of a numerical approximation. Global truncation error is the amount of truncation error that occurs in the
use of a numerical approximation to solve a problem.

Taylor’s Theorem and Approximations
The principal tool in the determination of truncation error is Taylor’s theorem, a key result in calculus that provides a formula for the amount of error in using a truncated Taylor series
to represent the corresponding function. Taylor’s theorem in its most general form applies to approximations of any degree; here we present only the information specifically needed to
analyze the error in Euler’s method.

Theorem 1 Let y be a function of one variable having a continuous second derivative on some
interval I = [0, tf ] and let f be a function of two variables having continuous first partial
derivatives on the rectangular region R = I × [y0 − r, y0 + s], with r, s > 0.

1. Given T and t in I, there exists some point τ ∈ I such that

.................. (1)
2. Given points (T, Y ) and (T, y) in R, there exists some point (T, η) ∈ R such that
f(T, y) = f(T, Y ) + fy(T, η)(y − Y ),................................. (2)
where fy is the partial derivative of f with respect to its second argument.

Theorem 1 serves to quantify the idea that the difference in function values for a smooth function should vanish as the evaluation points become closer.
One can be a little more restrictive when specifying the range of possible values for τ and η; however, nothing is gained by doing so. We cannot use Theorem 1 to compute the error in
an approximation. The theorem provides formulas for the error, but the catch is that there is no way to determine τ and η without knowing the exact solution. It may seem that this catch
makes the formulas useless, but this is not the case. We do know that τ and η are confined to a given closed interval, and therefore we can compute worst-case values for the quantities y"(τ )
and fy(T, η).
Example 1
Suppose we want to approximate the function ln(0.5 + t) near the point T = 0.5. We have
y(t) = ln(0.5 + t), y(T) = 0, y'(t) = 1 /(0.5 + t) , y'(T) = 1, y"(t) = −1/(0.5 + t)^2
.
Let’s assume that our goal4
is to find an upper bound for the largest possible error in approximating
y(t) by (t − 0.5) with 0 ≤ t ≤ 1. The approximation formula of Equation (1) yields

and the approximation error E is defined by
E = (t − 0.5) − y(t).
We therefore have

with 0 ≤ τ ≤ 1. Now let

Given the range of possible τ values, the worst case is |y"(τ )| = M; thus,
|E| ≤ 2(t − 0.5)2 , t ∈ [0, 1].

Local Truncation Error for Euler’s Method
Consider an initial value problem
y'= f(t, y(t)), y(0) = y0,
where f has continuous first partial derivatives on some region R defined by 0 ≤ t ≤ tf and
y0 − r ≤ y ≤ y0 + s. Euler’s method is a scheme for obtaining an approximate value yn+1 for

y(tn+1) using only the approximation yn for y(tn) and the function f that calculates the slope
of the solution curve through any point. Specifically, the method is defined by the formula
yn+1 = yn + hf(tn, yn), where h = tn+1 − tn.

We define the global truncation error at step n in any numerical approximation of (3) by
En = y(tn) - yn: .............. (5)
Our aim is to find a worst-case estimate of the global truncation error in the numerical scheme for Euler's method, assuming that the correct solution and numerical approximation stay within
R. This is a difficult task because we have so little to work with. We'll start by trying to determine the relationship between the error at time tn+1 and the error at time tn. Figure 1
shows the relationships among the relevant quantities involved in one step of the approximation. The approximation and exact solution at each of the two time steps are related by the error definitions. The approximations at the two time steps are related by Euler's method. We still need to have a relationship between the exact solution values at the two time steps, and this is
where Theorem 1 is needed.

The comparison of y(tn+1) and y(tn) must begin with Equation (1). Combining Equation (1), with T = tn, and Equation (4) yields
y(tn+1) = y(tn) + hf(tn; y(tn)) + (h^2/2)y"(t ): .............. (6)
This result is a step in the right direction, but it is not yet satisfactory. A useful comparison
of y(tn+1) with y(tn) can have terms consisting entirely of known quantities and error terms,
but not quantities, such as f(tn; y(tn)), that need to be evaluated at points that are not known
exactly. These quantities have to be approximated by known quantities. This is where Equation
(2) comes into the picture. Substituting
f(tn, y(tn)) = f(tn, yn) + fy(tn, η)[y(tn) − yn] = f(tn, yn) − fy(tn, η)En .into Equation (6) gives us
y(tn+1) = y(tn) + hf(tn, yn) − hfy(tn, η)En +(h^2/2)y"(τ ) ................... (7)
Equation (7) meets our needs because every term is either a quantity of interest, a quantity
that can be evaluated, or an error term. Subtracting this equation from the Euler formula (4)
yields
yn+1 − y(tn+1) = yn − y(tn) + hfy(tn, η)En −(h^2/2)y"(τ ),
or
En+1 = [1 + hfy(tn, η)]En −(h^2/2)y"(τ ) ................ (8)
This result indicates the relationship between the errors at successive steps. The local truncation error is defined to be the error in step n + 1 when there is no error in step n; hence,
the local truncation error for Euler’s method is (−h^2y"(τ ))/2. The local truncation error has
two factors of h, and we say that it is O(h^2) The quantity [1 + hfy(tn, η)]En represents the error at step n + 1 caused by the error at step n. This propagated error is larger than En if
fy > 0 and smaller than En if fy < 0. Having fy < 0 is generally a good thing because it causes truncation errors to diminish as they propagate...............

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