AIMS Lecture Notes 2006
PeterJ
.O
lver
11. Numerical Solution of
the Heat and Wave Equations
In this part, we study numerical solution methodss for the two most important equa
tions of onedimensional continuum dynamics. The
heat equation
models the diFusion of
thermal energy in a body; here, we treat the case of a onedimensional bar. The
wave
equation
describes vibrations and waves in continuous media, including sound waves, wa
ter waves, elastic waves, electromagnetic waves, and so on. ±or simplicity, we restrict our
attention to the case of waves in a onedimensional medium, e.g., a string, bar, or column
of air.
We begin with a general discussion of ²nite diFerence formulae for numerically ap
proximating derivatives of functions. The basic
fnite diFerence scheme
is obtained by
replacing the derivatives in the equation by the appropriate numerical diFerentiation for
mulae. However, there is no guarantee that the resulting numerical scheme will accurately
approximate the true solution, and further analysis is required to elicit bona ²de, conver
gent numerical algorithms. In dynamical problems, the ²nite diFerence schemes replace
the partial diFerential equation by an iterative linear matrix system, and the analysis of
convergence relies on the methods covered in Section 7.1.
We will only introduce the most basic algorithms, leaving more sophisticated variations
and extensions to a more thorough treatment, which can be found in numerical analysis
texts, e.g., [
5
,
7
,
28
].
11.1.
Finite Diferences.
In general, to approximate the derivative of a function at a point, say
f
0
(
x
)or
f
0
(
x
),
one constructs a suitable combination of sampled function values at nearby points. The
underlying formalism used to construct these approximation formulae is known as the
calculus o± fnite diFerences
. Its development has a long and inﬂuential history, dating
back to Newton. The resulting
fnite diFerence numerical methods
for solving diFerential
equations have extremely broad applicability, and can, with proper care, be adapted to
most problems that arise in mathematics and its many applications.
The simplest ²nite diFerence approximation is the ordinary
diFerence quotient
u
(
x
+
h
)
−
u
(
x
)
h
≈
u
0
(
x
)
,
(11
.
1)
4/20/07
186
c
±
2006
Peter J. Olver
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View Full DocumentOneSided Diference
Central Diference
Figure 11.1.
Finite Diference Approximations.
used to approximate the ±rst derivative o² the ²unction
u
(
x
). Indeed, i²
u
is diferentiable
at
x
,th
en
u
0
(
x
) is, by de±nition, the limit, as
h
→
0 o² the ±nite diference quotients.
Geometrically, the diference quotient equals the slope o² the secant line through the two
points
(
x, u
(
x
)
)
and
(
x
+
h, u
(
x
+
h
)
)
on the graph o² the ²unction. For small
h
i
s
should be a reasonably good approximation to the slope o² the tangent line,
u
0
(
x
), as
illustrated in the ±rst picture in Figure 11.1.
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 Fall '09
 Olver
 Algebra, Numerical Analysis, Equations, Peter J. Olver

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