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Chemical Engineering
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CH E374
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Joe Mmbaga
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Lecture

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University of Alberta
Department

Chemical Engineering

Course Code

CH E374

Professor

Joe Mmbaga

Description

CHE 374 Computational Methods in Engineering
MODULE 5: Numerical Differentiation & Integration
Differentiation basics
Definition of the derivative of a function f of x at x = x : 0
df f lim f f x 0 (1.1)
dx 0 xx0 x x 0
xx0
.
In numerical analysis, the difference x x x stays finite (does not really go to
0
zero). Derivatives are therefore approximated by differences:
f f f 0 (1.2)
0 x
.
An important theoretical result (known as mean-value theorem): in the interval
f f x 0
x0,x there is a point for which exactly f with generally
x x0
unknown.
Finite difference approximations
Suppose we know the function f at positions xi1 ix i 1hat are spaced by a distance
h We distinguish three approximations to the derivative at x x :
i
e
Forward difference: riat
f( ) xrede
df fi1 fi T
dx i x
x x x x
1 i- i i1
1 Backward difference:
tie
df fi fi1 f( ) x era
dx i x Tre d
x x
1 xi- x i i1
iatie
Central difference: f( ) x e der
Tr
df fi1 fi1
dx 2x
i
x
1 xi- x i x i1
From Figure 1c, we get the impression that the central difference approximation is
the most accurate. This we can ‘prove’ by means of Taylor expansions:
x2 x 3
fi1 f ixf i fi fiO x 4 (1.3)
2 6
2 3
f f xf x f x f O x 4 (1.4)
i1 i i 2 i 6 i
The O x 4 indicates that the leading order of the terms left out of the expansion is
4. Equation (1.3) can be rewritten such that
i1 fi x x2 3
fi fi i O x (1.5)
x 2 6
2 Thus
fi1 fi
i x O (1.6)
This means that the forward difference approximation of the first derivative has an
error proportional to x . In numerical jargon: the forward difference approximation
of the first derivative is first-order accurate (accurate to x to the first power).
This is an equation similar to the one obtained in Module 1, when we
dv
approximated the acceleration .
dt
Equation (1.4) can similarly be rewritten as
2
f fi fi1 x f x f O x 3 (1.7)
i x 2 i 6 i
thus
fi fi1
i O (1.8)
x
The backward difference approximation of the first derivative also has first-order
accuracy.
Subtracting Eq. (1.4) from Eq. (1.3) gives:
3
f f 2xf 2 x f O x 4 (1.9)
i1 i1 i 6 i
This can be rewritten as
2
i1 fi1 x 4
fi 2x 6 f iO x (1.10)
thus
i1 fi1 2
fi O (1.11)
2x
2
Apparently, the central approximation of the first derivative has second-order (h )
accuracy. For small x (which virtually always is the case), second-order accuracy
is much more accurate than first-order accuracy. Central difference being more
accurate than forward or backward difference could already be anticipated from the
figure.
3 The second derivative (and its accuracy) follows from adding up Equations (1.3)
and(1.4): (central derivative)
f f 2f h f O x 4 (1.12)
i1 i1 i i
thus
f 2f f
fi i1 i2 i O 2 (1.13)
x
The expressions given above for first and second-order derivatives and their level
of accuracy will largely cover your needs. Extensions can be made towards higher
derivatives, and higher-order accuracy.
Partial derivatives
Let f (x,y)be a function with two variables. If we keep y constant and differentiate
f (assuming f is differentiable) with respect to the variable , we obtain what is
called the partial derivative of f with respect to x which is denoted by
f
or fx
x
Similarly If we keep x constant and differentiate f (assuming f is differentiable)
with respect to the variable y, we obtain what is called the partial derivative of f
with respect to y which is denoted by
f
or fy
y
We might also define partial derivatives of function f as follows:
f fx x,y f x,
x0 O( ) (1.14)
x x
And with respect to y
f lim f ,y y f x y O( x) (1.15)
y y0 y
The above are forward differences. Similar expressions may be made for backward
and central approximations. The representation and calculation methods follow
similar approach as ordinary differential equations, but keeping in mind that we
have more than one dependent variable.
4 Second partial derivatives
Forward difference of the second partial derivative at a point i,j: O(x)
2
f f i2x,y 2fjxx,yif x ,y j i j
x 2 2 (1.16)
i, j
2 f ,y 2y 2f ,y y f x ,y
f i j i j i j (1.17)
y 2 i, j 2
Backward difference of the second partial derivative at a point i,j: O(x)
2 f ,y 2 x xy f x 2x,y
f i j i j i j (1.18)
x 2 i, j 2
f f iy j x ,y i jf x ,y y i j
2 2 (1.19)
y i, j
2
Central-difference approximation of the second partial derivative: O(x )
2
f f ix,y 2fj ,y i j i j (1.20)
x2 x 2
i, j
f f ,y y 2fx ,y f x , y
2 i j i 2j i j (1.21)
y i, j
5 Error propagation through differentiation
The figure below demonstrates a danger when numerically differentiating.
Smooth data noisy data
If the data contain noise, this noise is amplified when differentiating. Remember
cancellation errors introduced when two nearly equal numbers that contain small
errors are subtracted (subtractive cancellation).
6 Integration basics
There are two main reasons for you to need to do numerical integration: analytical
integration may be impossible or infeasible, or one may wish to integrate tabulated
data rather than known functions.
The problem of numerical integration is equivalent to evaluating the
general integral:
b
I(f ) f (x)d x (1.22)
a
In some cases we only know f (x)as a collection of points f f (x ). In other cases
i i
we know a closed expression for f (x) but the integral cannot be evaluated
analytically. In such cases we also base our numerical estimate of the integral on a
number of function evaluations i f (xi).
7 Let us define P (x) as a polynomial that satisfies f (x)at some
k
points x 0 x1, x2...x k. We then let:
I(f ) I(k ) (1.23)
We assume:
I(Pk) w0f (x0) w1f (1 ) ... k w k (x ) (1.24)
where the w aie weights. It can be shown that for equally spaced grid
points, and the weights being set equal to the Lagrange polynomials, that
the common integration formulas can be derived. Specifically, it can be
shown that for:
1
k 1 I(P k ba f (a) f (b) Trapazoidal rule
2
1 ab
k 2 I(P ) a f (a) 4f f (b) Simpson's rule
k 6 2
Newton-Cotes
First approach (midpoint rule).
The interval is divided in equal pieces of width h i1x i The integral is
estimated as the sum of all pieces:
I fxi1 i h (1.25)
2
8 Second approach (trapezoidal rule).
Inst

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