6Stiff equations
IB Numerical Analysis
6.1 Introduction
Initially, when people were developing numerical methods, people focused mostly
on quantitative properties like order and accuracy. We then develop many
different methods like multi-step methods and Runge-Kutta methods.
More recently, people started to look at structural properties. Often, equations
come with some special properties. For example, a differential equation describing
the motion of a particle would most probably conserve energy. When we
approximate it numerically, we would like the numerical approximation to satisfy
conservation of energy as well. This is what recent developments are looking at —
we want to look at whether numerical methods preserve certain nice properties.
We are not going to look at conservation of energy — this is too complicated
for a first course. Instead, we look at the following problem. Suppose we have a
system of ODEs for 0 ≤ t ≤ T :
y
0
(t) = f(t, y(t))
y(0) = y
0
.
Suppose T > 0 is arbitrary, and
lim
t→∞
y(t) = 0.
What restriction on
h
is necessary for a numerical method to satisfy
lim
n→∞
y
n
=
0
?
This question is still too complicated for us to tackle. It can only be easily
solved for linear problems, namely ODEs of the form
y
0
(t) = Ay(t),
for A ∈ R
N×N
.
Firstly, for what
A
do we have
y
(
t
)
→
0 as
t → ∞
? By some basic linear
algebra, we know this holds only if
Re
(
λ
)
<
0 for all eigenvalues
A
. To simplify
further, we consider the case where
y
0
(t) = λy(t), Re(λ) < 0.
It should be clear that if
A
is diagonalizable, then it can be reduced to multiple
instances of this case. Otherwise, we need to do some more work, but we’ll not
do that in this course.
And that, at last, is enough simplification.