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Setting
f
i
:=
→
f(x
i
)
, we obtain as
h
0,
f
i
+
1
−
f
i
f
(x
i
)
(δ
x
f)
i
+
O
=
+
O
(h)
=:
(h),
h
where
(δ
x
f)
i
is called the
one-sided difference quotient
of
f
with respect to
x
at
x
i
. The difference quotient is said to be
accurate of first order
since the remainder
term is
0. Analogous expressions hold for the time derivative
∂
t
.
Higher order finite differences allow obtaining approximations of order
O
(h)
as
h
→
(h
p
)
O
with
p
≥
2 rather than just
O
(h)
. If the function to be approximated has sufficient
regularity
,wehave
f
i
+
1
−
f
i
−
1
f
(x
i
)
(h
2
)
(h
2
),
C
3
(G),
=
+
O
=:
(δ
x
f)
i
+
O
for
f
∈
2
h
f
i
+
1
−
2
f
i
+
f
i
−
1
f
(x
i
)
(h
2
)
(δ
xx
f)
i
+
O
(h
2
),
C
4
(G).
=
+
O
=:
for
f
∈
h
2
With the difference quotients we turn next to the finite difference discretization of
the heat equation (
2.4
). Let
u
i
denote the approximate value of the solution
u
at
grid point
(t
m
,x
i
)
,i.e.
u
i
≈
u(t
m
,x
i
)
. For a parameter
θ
∈[
0
,
1
]
, we approximate
the partial differential operator
∂
t
u
−
∂
xx
u
at the grid point
(t
m
,x
i
)
by the finite
difference operator
(
1
u
m
+
1
i
u
i
−
m
i
−
θ)(δ
xx
u)
i
+
θ(δ
xx
u)
m
+
1
E
:=
−
i
k
,
(2.7)
and replace the partial differential equation (
2.4
)bythe
finite difference equations
(
1
u
m
+
1
i
θ)
u
i
+
1
−
2
u
i
u
i
−
1
θ
u
m
+
1
2
u
m
+
1
i
u
m
+
1
i
−
−
u
i
k
+
−
+
i
+
1
1
=
−
−
+
h
2
h
2
θf
m
+
1
i
m
i
θ)f
i
E
=
+
(
1
−
,i
=
1
,...,N,
m
=
0
,...,M
−
1
,
(2.8)
with initial conditions
u
i
1
,...,N
, and boundary conditions
u
k
=
u
0
(x
i
)
,
i
=
=
0,
0,
u
m
+
1
i
k
∈{
0
,N
+
1
}
,
m
=
0
,...,M
. We observe that for
θ
=
,
i
=
1
,...,N
,are
m
i
=
f
i
explicitly
in terms of
u
i
given in
E
, i.e. the scheme (
2.8
) is explicit. For
1, a linear system of equations must be solved for
u
m
+
1
i
θ
=
at each time step, i.e.
the scheme is
implicit
.
We write (
2.8
) in matrix form. To this end, we introduce the column vectors
u
m
(u
1
,...,u
N
)
,
f
m
(f
1
,...,f
N
)
,
(u
0
(x
1
),...,u
0
(x
N
))
,
=
=
0
=
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