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Vector Curl and Gradient, Lecture Notes - Mathematics - 13, Study notes of Mathematics
Time dependent problems and stability
Typology: Study notes
2010/2011
1 / 13
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Time-dependent problems Problems like the Diffusion Problem: Let
R
d
, d
or
∂u
∂t
k
u
f
in
×
[
, T
]
together with appropriate boundary conditons on
and
initial conditions at
t
are most often solved by the
method of lines
obtain weak form in space
approximate in space by finite elements
solve the resulting large system of ordinary differentialequations by some finite difference method usingtimesteps
t
Writing
u
h
(x
, t
n
∑ j
=
u
j
t
φ
j
(x) +
n
n
∂
j
=
n
u
j
t
φ
j
(x)
∈ H
1 E
we note that
∂u
h
∂t
(x
, t
n
∑ j
=
du
j
d
t
t
φ
j
(x) +
n
n
∂
j
=
n
du
j
d
t
t
φ
j
(x)
∈ H
1 E
For simplicity assume that
u
u
h
on
then
u
h
(x
, t
n
∑ j
=
u
j
t
φ
j
(x)
∂u
h
∂t
(x
, t
n
∑ j
=
du
j
d
t
t
φ
j
(x) =
n
j
=
u
j
t
φ
j
(x)
For the Galerkin equations
Ω
∂u
h
∂t
v
Ω
k
u
h
v
Ω
vf
The terms are the same as for the steady state(Poisson-like equation) we have considered so far in thiscourse except for the time derivative term
Ω
∂u
h
∂t
v
which using the above becomes
n
∑ j
=
u
j
t
Ω
φ
j
(x)
φ
i
(x)
when we take
v
φ
i
for each
i
,... , n
M
is also positive definite since
Ω
u
2 h
n
∑ j
=
n
∑ i
=
u
j
u
i
Ω
φ
j
φ
i
n
∑ i
=
u
i
n
∑ j
=
m
i,j
u
j
n
∑ i
=
u
i
M
u)
i
u
T
M
u
for every nonzero function in the approximation space.
The ordinary differential equation system got from Galerkinfinite element approximation in space is therefore
n
∑ j
=
u
j
t
Ω
φ
j
φ
i
n
∑ j
=
u
j
t
Ω
φ
j
φ
i
Ω
f φ
i
for
i
,... , n
. This is
n
coupled first order ordinary
differential equations.In other words, if we write
u(
t
) = (u
1
t
u
2
t
u
n
t
T
as a vector of the time dependent coefficients in the finiteelement expansion, these equations are
M
u(
t
A
u(
t
) = f
where
M
: mass matrix,
A
: stiffness matrix,
f
: load vector
M
u(
t
A
u(
t
) = f
Let
u
k
= (u
1
k
t
u
2
k
t
u
n
k
t
k
Eulers method
M
(u
k
u
k
t A
u
k
t
f
k
where
u
0
is given by the initial data.
In other words, the solution of the linear system
M
u
k
M
t A
)u
k
t
f
is required at each time step.Approximation properties are similar to the steady ellipticproblems; the main new issue here is
time-stability
: what
values of
t
allow an accurate solution to be computed.
Stability
: It is not just a question of how small
t
need to
be to give adequate accuracy, but the solution vectors
u
k
can tend to
as
k
if a choice is made which leads
to instability.If we can guarentee for example that the difference
w
k
between two sequences of solution vectors
u
k
and
v
k
say
satisfies
(w
k
T
M
(w
k
(w
k
T
M
(w
k
or
(w
k
T
T
(w
k
(w
k
T
T
(w
k
for any other positive definite matrix
T
(including the identity
matrix) then we will have a stable solution.
FEM – p.11/