Chapter 5 InitialValue Problems for Ordinary Differential Equations  Euler’s Method. Compute the approximations of y ( t ) at a set of ( usually equallyspaced ) mesh points a = t 0 < t 1 <…< t n = b . That is, to compute w i y ( t i ) = y i for i = 1, …, n.
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  Euler’s Method
Compute the approximations of y(t) at a set of ( usually equallyspaced ) mesh points a = t0< t1 <…< tn= b.
That is, to compute wi y(ti) = yi for i = 1, …, n.
5.2 Euler’s Method
Theorem: Suppose f is continuous and satisfies a Lipschitz condition with constant L on D = { (t, y)  a t b, – < y < } and that a constant M exists with y”(t) M for all a t b. Let y(t) denote the unique solution to the IVP y’ (t) = f(t, y), a t b, y(a) = , and w0, w1 , …wn be the approximations generated by Euler’s method for some positive integer n. Then for i = 0, 1, …, n
Difference equation
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  Euler’s Method
+ 0
+ i+1
Note: y”(t) can be computed without explicitly knowing y(t).
The roundoff error
Theorem: Let y(t) denote the unique solution to the IVP
y’ (t) = f(t, y), a t b, y(a) = ,
and w0, w1 , …wn be the approximations obtained using the above difference equations. If  i  < for i = 0, 1, …, n, then for each i
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
Definition: The difference method
w0 = ; wi+1 = wi + h( ti, wi), for each i = 0, 1, …, n – 1
has local truncation error
for each i = 0, 1, …, n – 1.
5.3 Higher Order Taylor Methods
Note: The local truncation error is just (yi+1wi+1)/h with the assumption that wi=yi.
The local truncation error of Euler’s method
Method of order 1
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
Taylor method of order n:
where
Note: Euler’s method can be derived by using Taylor’s expansion with n = 1 to approximate y(t).
Higher Order Taylor Methods
The local truncation error is O(hn) if y Cn+1[a, b].
4/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
T (2)(ti, wi) =
Taylor’s method of order 2:
Example: Apply Taylor’s method of order 2 and 4 to the IVP
y’ = y – t2 + 1, 0 t 2, y(0) = 0.5
Solution: Find the first 3 derivatives of f
f(t, y(t)) = y(t) – t2 + 1
HW:
p.271 #5 (a)(b)
f ’(t, y(t)) = y’(t) – 2t = y(t) – t2 + 1 – 2t
f ”(t, y(t)) = y’(t) – 2t – 2 = y(t) – t2– 2t – 1
f (3)(t, y(t)) = y’(t) – 2t – 2 = y(t) – t2– 2t – 1
Given n = 10, then
h = 0.2 and ti = 0.2i
Table 5.3 on p.269
wi+1 = 1.22wi – 0.0088i2– 0.008i + 0.22
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
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Hey! Isn’t the local truncation error
of Euler’s method ?
Seems that we can make a good use of it …
Other Euler’s Methods
Implicit Euler’s method
Usually wi+1 has to be solved iteratively, with an initial value given by the explicit method.
The local truncation error of the implicit Euler’s method
6/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
Trapezoidal Method
Two initial points are required to start
moving forward. Such a method is called
doublestep method. All the previously discussed
methods are singlestep methods.
Note: The local truncation error is indeed O(h2). However an implicit equation has to be solved iteratively.
Doublestep Method
Note: If we assume that wi – 1=yi– 1and wi=yi, the local truncation error is O(h2).
7/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
Method
Euler’s explicit
Euler’s implicit
Trapezoidal
Doublestep
Simple
Low order accuracy
stable
Low order accuracy
and time consuming
More accurate
Time consuming
More accurate,
and explicit
Requires one extra
initial point
Can’t you give me a method
with all the advantages yet without any
of the disadvantages?
Do you think it possible?
Well, call me greedy…
OK, let’s make it possible.
8/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  Higher Order Taylor Methods
Step 1:Predict a solution by the explicit Euler’s method
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Step 2: Correctwi+1 by Plugging it into the right hand side of the trapezoidal formula
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Modified Euler’s Method
Note: This kind of method is called the predictorcorrector method. This modified Euler’ method is a singlestep method of order 2. It is simpler than the implicit methods and is more stable that the explicit Euler’s method.
9/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  RungeKutta Methods
A singlestep method with highorder local truncation error without evaluating the derivatives of f.
In a singlestep method, a line segment is extended from (ti, wi) to reach the next point (ti+1, wi +1) according to some slope. We can improve the result by finding a better slope.
Idea
5.4 RungeKutta Methods
Must the slope
be the average of
K1and K2 ?
Check the modified Euler’s method:
Must the step size be h?
10/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  RungeKutta Methods
Generalize it to be:
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Step 2: Plug K2 into the first equation
Determine 1, 2, and p such that the method has local truncation error of order 2.
Step 1: Write the Taylor expansion of K2at ( ti , yi ) :
Step 3: Find 1, 2, and p such thati+1 = ( yi+1 – wi+1 )/h = O(h2).
11/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  RungeKutta Methods
Here are unknowns
and equations.
3
2
There are infinitely many solutions. A family of methods generated from these two equations is called RungeKutta method of order 2.
Note: The modified Euler’s method is only a special case of RungeKutta methods with p = 1 and 1 = 2 = 1/2.
Q: How to obtain higherordered accuracy?
12/14
Chapter 5 InitialValue Problems for Ordinary Differential Equations  RungeKutta Methods
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Classical RungeKutta Order 4 Method – the most popular one
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Chapter 5 InitialValue Problems for Ordinary Differential Equations  RungeKutta Methods
Note:
The main computational effort in applying the RungeKutta methods is the evaluation of f. Butcher has established the relationship between the number of evaluations per step and the order of the local truncation error :
evaluations per step
2
3
4
57
89
n10
Best possible LTE
HW:
p.280281
#1 (a), 10, 13
Since RungeKutta methods are based on Taylor’s expansion, y has to be sufficiently smooth to obtain better accuracy with higherorder methods. Usually the methods of lower order are used with smaller step size in preference to the higherorder methods using a large step size.
14/14