To solve differential equation problem:
If f is “smooth enough”, then
a solution will exist and be unique
we will be able to approximate it accurately with a wide variety of method
Two ways of expressing “smooth enough”
Lipschitz continuity
Smooth and uniformly monotone decreasing
7. Methods
To solve differential equation problem:
If f is “smooth enough”, then
a solution will exist and be unique
we will be able to approximate it accurately with a wide
variety of method
Two ways of expressing “smooth enough”
Lipschitz continuity
Smooth and uniformly monotone decreasing
7
12. 6.2 Euler’s Method
We have treated Euler’s method in Chapter 2.
There are two main derivations about Euler’s method
Geometric derivation
Analytic derivation
12
19. 6.3 Analysis of Euler’s Method
19
Proof: pp. 323-324 (You can study it by yourselves.)
O(h)
20. Theorem 6.4
Proof: pp. 324-325 (You can study it by yourselves.)
20
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21. Discussion
Both error theorems show that Euler’s method is only
first-order accurate (O(h)).
If f is only Lipschitz continuous, then the constants
multiplying the initial error and the mash parameter can
be quite large, rapidly growing.
If f is smooth and uniformly monotone decreasing in y,
then the constants in the error estimate are bounded for
all n.
21
22. Discussion
How is the initial error affected by f ?
If f is monotone decreasing in y, then the effect of the
initial error decreases rapidly as the computation
progresses.
If f is only Lipschitz continuous, then any initial error that
is made could be amplified to something exponentially
large.
22
23. 6.4 Variants of Euler’s
Method
Euler’s method is not the only or even the best scheme
for approximating solutions to initial value problems.
Several ideas can be considered based on some simple
extensions of one derivation of Euler’s method.
23
24. Variants of Euler’s Method
24
We start with the differential equation
And replace the derivative with the simple difference quotient derived in
(2.1)
What happens if we use other approximations to the derivative?
25. Variants of Euler’s Method
If we use
then we get the backward Euler method
If we use
then we get the midpoint method
25
O(h)
O(h2
)
Method 1
Method 2
26. Variants of Euler’s Method
If we use the methods based on interpolation (Section 4.5)
then we get two numerical methods
26
O(h2
)
O(h2
)
Method 4
Method 3
27. Variants of Euler’s Method
By integrating the differential equation:
(6.23)
and apply the trapezoid rule to (6.23) to get
Thus
(6.25)
27
Method 5
O(h3
)
28. Variants of Euler’s Method
We can use a midpoint rule approximation to integrating (6.23)
and get
28
O(h3
)
Method 6
29. Discussion
What about these method? Are any of them
any good?
Observations
Methods 2, 3, and 4 are all based on derivative
approximations that are O(h2
), thus they are more
accurate than Euler method and method 1 (O(h)).
Similarly, methods 5 and 6 are also more
accurate.
Methods 2, 3, and 4 are not single-step methods,
but multistep methods. They depend on
information from more than one previous
approximate value of the unknown function.
29
30. Discussion
Observations (con.)
Concerning methods 1, 4, and 5, all of these formulas involve we
cannot explicitly solve for the new approximate values Thus these
methods are called implicit methods.
Methods 2 and 3 are called explicit methods.
30
36. 6.4.2 Implicit Methods and Predictor-
Corrector Schemes
How to get the value of yn+1? Using Newton’s method or
the secant method or a fixed point iteration.
36
Let y = yn+1
42. Discussion
Generally speaking, unless the differential equation is
very sensitive to changes in the data, a simple
predictor-corrector method will be just as good as the
more time-consuming process of solving for the exact
values of yn+1 that satisfies the implicit recursion.
42
43. Discussion
If the differential equation is linear, we can entirely
avoid the problem of implicitness.
Write the general linear ODE as
43
44. 6.5 Single-step Method: Runge-Kutta
The Runge-Kutta family of methods is one of the most
popular families of accurate solvers for initial value
problems.
44