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## Review of numerical methods for ODEs Numerical Methods for PDEs Spring 2007

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### Forward Euler blows up. We say ... Forward Euler blows up. ... The blow-up of forward Euler is due to the numerical method for this IVP, not the IVP itself. ... – PowerPoint PPT presentation

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Title: Review of numerical methods for ODEs Numerical Methods for PDEs Spring 2007

1
Review of numerical methods
for ODEs Numerical Methods for
PDEs Spring 2007
• Jim E. Jones
• References
• Numerical Analysis, Burden Faires
• Scientific Computing An Introductory Survey,
Heath

2
Ordinary Differential Equation Initial Value
Problem (IVP)
• Numerical Solution rather than finding an
analytic solution for y(t), we look for an
approximate discrete solution wi (i0,1,,n) with
wi approximating y(aih)

t
tb
ta
h
3
Forward Euler via Taylors Theorem
Write down a Taylor expansion for y about ti
Evaluate at ti1 and note yf
Drop h2 term to get Eulers method (forward Euler)
4
Backward Euler via Taylors Theorem
Write down a Taylor expansion for y about ti1
Evaluate at ti and note yf
Drop h2 term to get Eulers method (backward
Euler)
5
Explicit vs. Implicit Methods
Backward Euler
Forward Euler
• Forward Euler is an explicit method. Computing
the new approximation wi1 is simple.
• Backward Euler is an implicit method. Computing
the new approximation wi1 can be more
complicated it appears on both sides of the
equation and inside the f function.

6
Ordinary Differential Equation Example 1
Forward Euler with h0.1
Backward Euler with h0.1
Solving for w1 using Newtons Method
More examples at http//www.cse.uiuc.edu/iem/ode/
7
Ordinary Differential Equation Example 2.a
Forward Euler with h0.1
Backward Euler with h0.1
Both computed solutions go to zero as t increases
like the true ODE solution
8
Ordinary Differential Equation Example 2.b
Forward Euler with h1.1
Backward Euler with h1.1
Backward Euler go to zero as t increases. Forward
Euler blows up.
9
Ordinary Differential Equation Example 2.b
Forward Euler with h1.1
We say that forward Euler is unstable. Stability
is an important property of numerical methods for
ODEs and PDEs. Our goal over the next few
lectures will be to explain and quantify this
behavior.
Backward Euler with h1.1
Backward Euler go to zero as t increases. Forward
Euler blows up.
10
Ordinary Differential Equation Example 2.b
Forward Euler with h1.1
To do this investigation, we will need some
results from the theory of ODEs. These results
are from Burden Faires.
Backward Euler with h1.1
Backward Euler go to zero as t increases. Forward
Euler blows up.
11
Lipschitz condition definition
• A function f(t,y) satisfies a Lipschitz
condition in the variable y on a set D in R2 if a
constant L gt 0 exists with
• whenever (t,y0) and (t,y1) are in D. The
constant L is called a Lipschitz constant for f.

12
Convex set definition
• A set D in R2 is said to be convex if whenever
(t0,y0) and (t1,y1) belong to D and l is in
0,1, the point
• ((1-l)t0lt1, (1-l)y0ly1) also belongs to D.

convex
not convex
13
Theorem bounded partial derivative a Lipschitz
• Suppose f(t,y) is defined on a convex set D in
R2. If L is a bound on fy, i.e.
• then f satisfies a Lipschitz condition on D in
the variable y with Lipschitz constant L.

14
Theorem Lipschitz a Existence and Uniqueness
• Suppose that
• and that f(t,y) is continuous on D and satisfies
a Lipschitz condition on D in the variable y,
then the IVP
• has a unique solution y(t) for t in a,b.

15
Well posed IVP definition
• The IVP
• is a well-posed problem if
• A unique solution y(t) exists, and
• There exists constants e0 gt0 and K gt 0 such that
for any e in (0,e0), whenever d(t) is continuous
with d(t) lt e for all t in a,b, and when d0
lt e, the IVP
• has a unique solution z(t) satisfying

16
Well posed IVP definition
• The IVP
• is a well-posed problem if
• A unique solution y(t) exists, and
• There exists constants e0 gt0 and K gt 0 such that
for any e in (0,e0), whenever d(t) is continuous
with d(t) lt e for all t in a,b, and when d0
lt e, the IVP
• has a unique solution z(t) satisfying

Original IVP
Perturbed IVP
Difference in solutions to 2 IVPs
17
Characteristics of well-posed problems
• In words, an IVP is well-posed if small
perturbations in the data (the right-hand side
and initial condition) produce correspondingly
small changes in the solution.
• When solving an IVP numerically, we can not
expect reasonable results for problems that are
not well-posed (Why?). Such problems are called
ill-posed.

18
Characteristics of well-posed problems
Numerical solution will always be dealing with
perturbed problems due to round-off error in
computer representation. If these small errors
produce large changes in the solution, we cant
expect good agreement between the computed
solution and the solution to the IVP
• In words, an IVP is well-posed if small
perturbations in the data (the right-hand side
and initial condition) produce correspondingly
small changes in the solution.
• When solving an IVP numerically, we can not
expect reasonable results for problems that are
not well-posed (Why?). Such problems are called
ill-posed.

19
Theorem Lipschitz a Well-posed
• Suppose that
• and that f(t,y) is continuous on D and satisfies
a Lipschitz condition on D in the variable y,
then the IVP
• is well-posed.

20
Is the Example 2.b well-posed?
Recall the Lipschitz condition or
The rhs f(t,y)-2y clearly satisfies the
Lipschitz condition with L2. So the IVP is
well-posed. The blow-up of forward Euler is due
to the numerical method for this IVP, not the IVP
itself.
21
Is the Example 2.b well-posed?
To investigate the problem were seeing with
forward Euler, we will need to define some
properties for numerical methods namely,
consistency, stability, and accuracy (or
convergence). Again Burden Faires or Heath are
good references.
Recall the Lipschitz condition or
The rhs f(t,y)-2y clearly satisfies the
Lipschitz condition with L2. So the IVP is
well-posed. The blow-up of forward Euler is due
to the numerical method for this IVP, not the IVP
itself.
22
Local truncation error definition
• The difference method
• has local truncation error
• for each i0,1,,n-1

23
Local truncation error definition
• The difference method
• has local truncation error
• for each i0,1,,n-1

The local truncation error is a measure of the
degree to which the true IVP solution fails to
satisfy the difference equation.
24
Local truncation error definition
• The difference method
• has local truncation error
• for each i0,1,,n-1

The local truncation error is error in single
step, assuming the previous step is exact, scaled
by the mesh size h.
25
Local truncation error for Euler
• From the Taylor theorem derivation for forward
• so
• so if y is bounded, the local truncation error
is O(h). Same result holds for backward Euler.

26
Definition of consistency and convergence
• A difference method is consistent with the
differential equation if
• A difference method is convergent (or accurate)
with respect to the differential equation if

27
Definition of consistency and convergence
• A difference method is consistent with the
differential equation if
• A difference method is convergent (or accurate)
with respect to the differential equation if

The missing piece of the puzzle is the concept of
stability of a numerical method. Next time well
define stability and see that Consistency
Stability Accuracy