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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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