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CE 341/441 - Review 2 - Fall 2004
p. R2.
REVIEW NO. 2NUMERICAL DIFFERENTIATION • Find a
discrete
approximation to differentiation
- Use numerical differentiation to solve o.d.e.’s and p.d.e.’s on a computer
- Recall that a computer doesn’t do differential/integral mathematics and only deals
with discrete functional values
Generic Method to Derive a Difference Formula:
where
= functional values at nodes
= coefficients of the formula being derived
when
nodes are used (or better for some central approximations)
f
i
p (
)
E
a
α
f
α
a
β
f
β
a
λ
f
λ
h
p
f
α
f
β
f
λ
a
α
a
β
a
λ
E
O h
N
p
N
CE 341/441 - Review 2 - Fall 2004
p. R2.
- Procedure:
- Substitute Taylor Series expansions for
etc. about node
i
- Rearrange equations such that coefficients multiply equal order derivatives at node
i
and generate algebraic equations by setting coefficients of
equal to 1 and the
other coefficients equal to zero
etc.
Numerical Differentiation Formulae Using Interpolating Polynomials: • Use at least
nodes with the interpolation formula to establish an approximation to
the
p
th
derivative
- Any interpolating technique/formula can be used• The numerical differencing formula is simply the differentiated interpolating polyno-
mial evaluated at one of the nodes used for interpolation
- The error can be computed based on the error of the interpolating formula
f
α
f
β
f
i
p (
)
p
N
a
α
a
β
p
CE 341/441 - Review 2 - Fall 2004
p. R2.
- Error can be derived by
- Applying Taylor series expansions to the terms in the differentiation formula• Differentiating the error in the interpolation function (assuming that this error is
expressed as a series and does not depend on
- Higher order accuracy leads to better answers for larger
(presumably less work for
computer for a given level of accuracy)
- However, higher order accuracy is not
always
better
- Recall that for interpolation, piecewise linear was sometimes better than high order
interpolation
- The same holds true for differentiation. It depends on the function• Also there are trade-offs in order of accuracy versus actual implementation cost on a
computer
ξ
h
CE 341/441 - Review 2 - Fall 2004
p. R2.
INTRODUCTION TO O.D.E.’s AND P.D.E.’SPartial Differentiation • Simply apply numerical differentiation formulae relative to the independent variable/
direction in which you are differentiating while holding indices associated with all otherindependent variables/directions constant.
Solving Single Equation O.D.E. I.V.P.’s • Solve
with a specified i.c.
- Apply the Euler method by approximating the d.e. at
and applying a forward
difference approximation for the first derivative
⇒
- Simply advance from one time level to the next by substituting known values
into
the right hand side and solving for
→
“time marching”
dy ----- dt
f
y t
y t
o
(
y
o
t
t
j
y
j
1
y
j
t
f
y
j
t
j
,
y
j
1 +
y
j
t f
y
j
t
j
,
y
j
t
j
,
y
j
1
CE 341/441 - Review 2 - Fall 2004
p. R2.
Solutions to P.D.E.’s • Space-time dependent p.d.e.’s are solved by defining a space-time grid of discrete nodes^ INSERT FIGURE NO. 68 • Discrete solutions are advanced in time by writing a sufficient number of algebraic
equations to allow the solution of all unknowns at a given time level
→
“time marching”
- Solution is found at a specific time level prior to advancing to the next time level• Finite difference approximations are used to discretize the p.d.e.’s at either the
known or unknown time levels.
t
j+
j
j-
2 1
j=
(i,j)
i=
1
2
i-
i
i+
n-
n
i=n+
x
∆
t
∆
x
known valuesunknown values
CE 341/441 - Review 2 - Fall 2004
p. R2.
- Time discretizations for first time derivatives
- Explicit: approximate the time derivative at the known time level j using a first order
forward approximation in time and evaluate all spatial derivatives at the known timelevel j
- Implicit: approximate the time derivative at the unknown time level
using a
backward approximation in time and evaluate all spatial derivatives at the unknowntime level
- Crank-Nicolson: approximate the time derivative at an intermediate time level
using a second order central approximation in time and evaluate the spatial
derivatives at the intermediate time level
(applying linear interpolation to
express the intermediate node values at
in terms of full node values at
and
- Spatial differentiation is typically implemented using central finite difference approxi-
mations
- Explicit methods do not lead to systems of simultaneous equations since there is no
coupling in the discrete equations between nodes at the unknown time level.
- Implicit and Crank-Nicolson methods do lead to systems of simultaneous equations
since there is coupling in the discrete equations between nodes at the unknown timelevel.
j
j
j
j
j
j
j
CE 341/441 - Review 2 - Fall 2004
p. R2.
- Steps to derive integration formulae
- Define a sub-interval• Define interpolation/integration points over the sub-interval as well as the interpola-
tion function type (and order)
(e.g. Lagrange or Hermite)
or
- Integrate the approximating interpolating function over the sub-interval• Now sum up integrals over all sub-intervals to develop the “extended formulae”
Newton Cotes Closed Formulae • Derived by using equispaced interpolation points and Lagrange interpolation• Integration points are always at the end points of the sub-interval
- Trapezoidal Rule and Simpson’s
Rule
Newton Cotes Open Formulae • Same as the closed formulae except that the sub-interval now extends beyond interpola-
tion (or integration) points
g x
f
i
V
i
x (
i^
0
N =
g x
f
i
α
i^
x (
i^
0
N =
f
i
(^1) ( (^)
)
i
β
i^
x (
i^
0 N =
CE 341/441 - Review 2 - Fall 2004
p. R2.
Gauss-Legendre Integration Formulae • Derived by using non-equispaced interpolation points and Hermite interpolation• Specifically the integration points are selected such that the integrals of the
func-
tions (associated with the derivative terms) are equal to zero.
- Thus • Now we select the integration points
such that
β
i^
x (
I
g x
x d
1
+1^ ∫ –
g x
α
i^
x (
f
i
i
0
N =
β
i^
x (
f
i
(^1) ( (^)
)
i^
0
N =
I
f
i
α
i
x (
x d
1
+1^ ∫ –
i
0 N =
f
i
(^1) ( (^)
)
β
i^
x (
x d
1
+1^ ∫ –
i^
0 N =
x
i
β
i^
x (
x d
1
+1^ ∫ –
CE 341/441 - Review 2 - Fall 2004
p. R2.
Methods for Evaluating the Accuracy of Integration Methods For a Sub-interval Method 1: Develop Taylor Series expansions for f(x) and the functional values at thenodes • Equation for Simpson’s 1/3 rule
where
= the exact integral
= the integration formula
INSERT FIGURE NO. 110
E
f
x (
x
h --- 3
d
2 0
h
f
o
f
^1
f
^2
f
x (
x d
0 2
h
∫ h --- 3
f
o
f
^1
f
^2
x
0
=
x
f
0
f
1
f
2
x
1
=h
x
2
=2h
CE 341/441 - Review 2 - Fall 2004
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- Develop Taylor Series expansions for
and
about the midpoint of the
interval
- Substitute in and integrate Method 2: Use error terms from the interpolating function • Apply the error in the interpolation function in series form (using the first few terms)•
DO NOT
use
based on evaluations of the last derivative at
where
is a point in
the interval since
⇒
⇒
f
x (
f
o
f
^1
f
^2
x
1
h
e x
ξ
ξ
ξ
ξ
x (
e x
f
x (
g x
E
f
x (
x
g x
x d ∫ – d ∫ = E
e x
x d
CE 341/441 - Review 2 - Fall 2004
p. R2.
⇒
- We note that • Substituting
remains constant no matter what sub-interval spacing
you choose
remains approximately constant over the interval
no matter what sub-
interval
you choose.
N
b
a
h
b
a
E
C b
a
h
n
1
1 --- N
f
m (
)
x
i
(
i^
1
N =
1 --- N
f
m (
)
x
i
(
i^
1
N =
f
m (
)
x
i
(
E
C b
a
h
n
1
f
m (
)
x
i
b
a
h
f
m (
)
x
i
(
a b
[
]
h
CE 341/441 - Review 2 - Fall 2004
p. R2.
INSERT FIGURE NO. 112
log E
log h
1
h
n- n-
CE 341/441 - Review 2 - Fall 2004
p. R2.
- Evaluate the integral using two or more spacings. This allows us to evaluate
etc.
with
with
- Solve for• Improve accuracy by two orders
etc.
- Improve accuracy by four orders!
I C D
˜ I
h
h
˜ I
2
h
h
I C
˜ I
h
˜ I
2
h
˜ I
4
h
I C D
