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Lecture notes on euler's method, a numerical approach to solving ordinary differential equations (odes). The notes cover the derivation, basic analysis, and improvements to euler's method, including taylor series methods and multi-point methods. The document also includes examples and exercises.
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Euler’s Method
Improving Euler’s Method
Lecture Notes #3 — Euler’s Method
Peter Blomgren,
Department of Mathematics and Statistics
Dynamical Systems Group
Computational Sciences Research Center
San Diego State University
San Diego, CA 92182-
http://terminus.sdsu.edu/
Spring 2009
Euler’s Method
Improving Euler’s Method
Outline
(^1) Euler’s Method
Example
Quantifiable Properties
Derivation, and Basic Analysis
2 Improving Euler’s Method
Taylor Series Methods
Multi-Point Methods
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Euler’s Method — Example, y
′ = y + 2t − 1; y (0) = 1
Exact Solution: y (t) = 2e
t − 2 t − 1
Euler’s method on the interval [0, 1], with
h ∈ { 1 / 2 , 1 / 4 , 1 / 8 , 1 / 16 }.
0 0.2 0.4 (^) 0.6 0.8 1
1
2
Exact Solution h=1/ h=1/ h=1/ h=1/
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Euler’s Method — Things to Quantify
Accuracy:
We have seen that the quality of the numerical solution
depends on the step size h.
Some of the concepts we need to define in order to analyze
numerical methods for ODEs:
Consistency:
Is the numerical scheme solving the right problem?
Stability:
Is the numerical scheme robust with respect to propagation
of round-off errors?
Convergence:
Do we get the right numerical solution as h → 0???
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Consistency
Euler’s method
y i+ = y i
is consistent with the differential equation
y
′ (t) = f (t)
since the Local Truncation Error satisfies
lim
h→ 0
Euler (h) = lim
h→ 0
h
y
′′ (ξ i
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Accuracy
A method is said to be of order p if
lim
h→ 0
LTE(h)
h
p
and
lim
h→ 0
LTE(h)
h
p+ǫ
= ±∞, ǫ > 0.
Since LTE Euler (h) =
h
y
′′ (ξ i ), p Euler
Euler’s Method is a first order method.
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Region of Stability
−2 −
Euler’s method is stable only if |1 + hλ| ≤ 1. That is, hλ must be
inside the disk of radius 1, centered at −1 in the complex plane.
If λ is real hλ must be in the interval [− 2 , 0]
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Stability: Example
Consider the ODE (exact solution y (t) = e
− 20 t )
y
′ (t) = − 20 y (t), y (0) = 1
Since λ = −20, we must have h < 0 .1 for stability...
0 0.2 0.4 0.6 0.8 1
0
Exact Solution h=0.
0 0.2 0.4 0.6 0.8 1
0
Exact Solution h=0.
0 0.2 0.4 0.6 0.8 1
0
Exact Solution h=0.
0 0.2 0.4 0.6 0.8 1
0
2
Exact Solution h=0.
h = 0. 001 h = 0. 05 h = 0. 11 h = 0. 11
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Summary: Key Concepts Introduced
Local Truncation Error, LTE(h)
The local error introduced by the discretization.
Accuracy
The order of accuracy is the largest integer p such that
lim
h→ 0
LTE(h)
h
p
Consistency
A method is consistent if
lim
h→ 0
LTE(h) = 0
Euler’s Method
Improving Euler’s Method
Example
Quantifiable Properties Derivation, and Basic Analysis
Summary: Key Concepts Introduced, II
Stability
A scheme is unstable if it produces exponentially growing
solutions for a problem for which the exact solution is bounded.
Usually stability introduces restrictions on the step size h.
Region of Stability
The range of hλ for which the selected method is stable.
Convergence
The numerical solution converges to the exact solution if the
scheme is Consistent and Stable.
Euler’s Method
Improving Euler’s Method
Taylor Series Methods
Multi-Point Methods
Beyond Euler’s Method
Euler’s method is easy to implement, but...
The step-size h must be very small to achieve an acceptable
level of accuracy (locally, the LTE).
If we are solving over a long time period [0, T ] with small
step-size, the method is expensive (requires many iterations)
and slow.
Local errors accumulate. The LTE ∼ O(h) but we need ∼ 1 /h
iterations, in order to compute up to a fixed final time T.
This could mean trouble?
Euler’s Method
Improving Euler’s Method
Taylor Series Methods
Multi-Point Methods
Back to the Drawing Board — More Taylor Series...
Our first improvement of Euler’s method is to keep more terms in
the Taylor expansion
y (t i+
n ∑
k=
h
k
k!
y
(k) (t i
h
n+
(n + 1)!
y
(n+1) (ξ i ), ξ i ∈ [t i , t i+
The last term is the remainder term which corresponds to the
local truncation error. Recall that for Euler’s method we set
n = 1, and ignored higher order terms.
From the differential equation
y
′ (t) = f (t, y ), y (t 0 ) = y 0
we can get expressions for higher order derivatives of y with the
help of the chain rule.
Euler’s Method
Improving Euler’s Method
Taylor Series Methods
Multi-Point Methods
Example: Higher Order Taylor Series Methods, I
We consider
y
′ (t) = y (t) + 2t − 1 , y (0) = 1.
We get
f (t, y ) = y + 2t − 1
f
′ (t, y ) = 2 + 1 · (y + 2t − 1) = y + 2t + 1
f
′′ (t, y ) = 2 + 1 · (y + 2t − 1) = y + 2t + 1
f
(n) (t, y ) = y + 2t + 1
Euler’s Method
Improving Euler’s Method
Taylor Series Methods
Multi-Point Methods
Example: Higher Order Taylor Series Methods, II
0 0.2 0.4 0.6 0.8 1
1e-
1e-
1e-
1e-
1st order
2nd order
3rd order 4th order
Figure: The error (on a logarithmic scale) for 1st, 2nd, 3rd and 4th
order Taylor methods applied to y
′ = y + 2t − 1 , y (0) = 1 on the
interval [0, 1] with step size h = 0.1.
