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Applied Mathematics 4301:
Numerical Methods for PDEs
Lecture #13:
Introduction to numerical methods for
hyperbolic PDEs
Wed aft. 4:106:40
S. W. Mudd Bldg. 1024
Prof. David Keyes, instructor
S. W. Mudd Bldg. 215
[email protected]
Yan Yan, teaching assistant
[email protected]
Lecture #13 (Part B)
3 December 2008
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View Full Document Plan of presentation
• Review properties of hyperbolic equations, and
compare and contrast to our wellstudied parabolic
– Classical secondorder hyperbolic shares the spatial
analysis of the Laplacian, but has different timelike
term (makes it timereversible)
– Classical firstorder hyperbolic is a subset of the general
parabolic problem, without the regularizing Laplacian
term (here pure advection, the nonselfadjoint part)
xx
tt
u
a
u
2
=
x
t
au
u
±
=
Pedagogical philosophy
•
We will “spiral staircase” through the material
•
Each sector of the staircase is subject matter (equation type,
discretization method, solution algorithm, application set, etc.)
•
Each time we reach a new level with one sector, we are ready to
revisit the next sector on a higher level of sophistication
•
Good chance for review
☺
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View Full Document From whence do 1
st
order equations come?
•F
l
u
x
f(u)
is
au
•
Conservation of
u
in a fixed
interval
[
x
1
,x
2
]
•
For sufficiently smooth
functions
•
Interchanging limits
•S
i
n
c
e
[
x
1
,x
2
]
is arbitrary, the
integrand must vanish
identically everywhere
•
If velocity is constant
))
,
(
(
))
,
(
(
)
,
(
2
1
2
1
t
x
u
f
t
x
u
f
dx
t
x
u
dt
d
x
x
−
=
∫
∫
∫
∂
∂
−
=
2
1
2
1
))
,
(
(
)
,
(
x
x
x
x
dx
t
x
u
f
x
dx
t
x
u
dt
d
0
))]
,
(
(
)
,
(
[
2
1
=
∂
∂
+
∂
∂
∫
x
x
dx
t
x
u
f
x
t
x
u
t
Given a passive scalar
u(x,t)
(e.g., thermal energy) being convected in a steady 1D
medium streaming with velocity
a(x,t),
positive to the right
0
))
,
(
)
,
(
(
)
,
(
=
∂
∂
+
∂
∂
t
x
u
t
x
a
x
t
x
u
t
0
=
+
x
t
au
u
Advection equation is a “singularly
perturbed” limit of the parabolic case
)
,
(
)
,
(
)
,
(
)
)
,
(
(
t
x
f
u
t
x
r
x
u
t
x
a
x
u
t
x
d
x
t
u
+
+
∂
∂
+
∂
∂
∂
∂
=
∂
∂
• Diffusion coefficient
d(x,t)
is always nonnegative
• Advection coefficient
a(x,t)
may be of either sign
•I
f

a
>>
d
, the equation is dominated by advection, with
diffusion arising only in a boundary layer
• We have studied the steady form of this equation in the unit
on elliptic equations
• Special attention is required (upwinding, PetrovGalerkin)
due to nonselfadjointness
• When
a(x,t)
is a function of
u
(nonlinear), further attention is
required; see lecture #11.
• Reducing the regularizing part
d(x,t)
makes things worse
still – but delightfully interesting
☺
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View Full Document From whence do 2
nd
order equations come?
•D
e
n
s
i
t
y
ρ
,
velocity
u,
pressure
p,
sound speed
a
•
Conservation of mass
•
Conservation of momentum
•
Adiabatic equation of state
•
Crossdifferentiate both PDEs
•
Substitute second PDE in first
•
Substitute from state eqn
x
t
p
u
−
=
)
(
Given a pair of such coupled firstorder advection equations, we can derive a
single secondorder equation. Considering only the dominant terms (acoustics):
0
)
(
=
+
x
t
u
d
a
dp
2
=
,
0
)
(
=
+
xt
tt
u
xx
tx
p
u
−
=
)
(
,
0
=
−
xx
tt
p
0
2
=
−
xx
tt
a
Canonical 1
st
order hyperbolic
•D
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This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.
 Fall '08
 Keyes

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