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Fig. 2.1
Time-space grid
2.3 Numerical Methods for the Heat Equation
Let the space domain
G
=
(a, b)
⊂ R
be an open interval and let the time domain
:=
J
(
0
,T)
for
T>
0. Consider the initial-boundary value problem:
Find
u
:
J
×
G
→ R
such that
∂
t
u
−
∂
xx
u
=
f(t,x),
in
J
×
G,
(2.4)
u(t, x)
=
0
,
on
J
×
∂G,
=
u(
0
,x)
u
0
,
in
G,
where
u(
0
,x)
0 on the boundary is called
the homogeneous Dirichlet boundary condition
. We explain two numerical methods
to find approximations to the solution
u(t, x)
of the problem (
2.4
). We start with the
finite difference method.
=
u
0
is the
initial condition
and
u(t, x)
=
2.3.1 Finite Difference Method
In the finite difference discretization, the domain
J
G
is replaced by discrete grid
points
(t
m
,x
i
)
and the partial derivatives in (
2.4
) are approximated by difference
quotients at the grid points. Let the
space grid points
be given by
×
x
i
=
a
+
ih, i
=
0
,
1
,...,N
+
1
,h
:=
(b
−
a)/(N
+
1
)
=
x,
(2.5)
which are equidistant with mesh width
h
, and the
time levels
by
t
m
=
mk, m
=
0
,
1
,...,M, k
:=
T/M
=
t.
(2.6)
The time-space grid is illustrated in Fig.
2.1
.
Assume that
f
∈
C
2
(G)
. Then, using Taylor's formula, we have
f(x
+
h)
−
f(x)
h
2
f
(ξ ),
f
(x)
=
−
ξ
∈
(x, x
+
h).
h
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