Chapter 2
CLASSIFICATION
OF
PARTIAL
DIFFERENTIAL EQUATIONS (PDEs)
Classification of PDEs
Physical Classification
Equilibrium
Problems
Mathematical Classification
Marching
Problems
Hyperbolic
Parabolic
Elliptic
PDEs
PDEs
PDEs
2-1 Physical Classification
(a) Equilibrium Problems
Closed Domain + Boundary Conditions
BCs must be satisfied on B
B
D
PDEs must be satisfied in D
2
==> B.V.P., e.q. ∇ T = 0
2
∇ ψ = 0
Page 2-1
2
∇ P = f ( u, v, ∇•u )
(b) Marching Problems (One-Way Problems)
Open Domain (time or time like) + I. C. and/or B. C.
e.g.1: Unsteady Heat Conduction Problems
2
∂T
= α∇ T
∂t
t
t
B.C.
B.C
PDE
I. C.
.
x
e.g.2: Boundary Layer Flow Without Separation
y
∂u ∂v
+
= 0
∂x ∂y
u∞
2
X
==> Marching in x-direction
==> y-direction?
==> Initial-boundary value problem
==> What if separation?
Page 2-2
∂ u
∂u
∂u
= ν
u +v
∂x
∂y
∂y2
e.g.3: Pure Initial Value Problem
Linear/Nonlinear Bergers Equation
∂u
∂u
= –a
∂t
∂x
I.C.
∂u
∂u
= –u
∂t
∂x
u ( x, 0 ) = f ( x )
Wave Equation
2
2
∂ u
∂ u
= a2
∂t2
∂x2
I.C.
u ( x, 0 ) = f ( x )
u t ( x, 0 ) = g ( x )
2-2 Mathematical Classication
(a) Linear and Nonlinear PDEs
Consider the following general 2nd-order PDE
AΦ xx + BΦ xy + CΦ yy + DΦ x + EΦ y + FΦ + G = 0
(1-1)
The characteristic differential eq. of (1-1) is
dy 2
dy
A  – B  – C = 0
 d x
 d x
Page 2-3
(1-2)
Gives
± B 2 – 4AC
 dy = B
------------------------------------- d x
2A
⇒
(1-3)
The Discriminant of (1-1) is
D
= B 2 – 4AC
(1-4)
==> if D > 0 at pt (x0,y0) => 2 characteristic curves => Hy at pt (x0,y0)
==> if D = 0 at pt (x0,y0) => 1 characteristic curve => Pa at pt (x0,y0)
==> if D < 0 at pt (x0,y0) => 0 characteristic curve => Ell at pt (x0,y0)
==> if D changes sign => Mixed type
e.g., Transonic Flows, e.g., 2D Potential flow eq.
D
( 1 – M 2 )Φ xx + Φ yy = 0
= – 4 ( 1 – M2 )
e.g.,
u tt = a 2 u xx
A = a2
I.C. => u ( x, 0 ) = f ( x ) and u t ( x, 0 ) = g ( x )
with
B = 0
C = –1
D
⇒
= 4a 2
On the x-t plane, one can show that
dt
1
= ± --dx
a
⇒
Page 2-4
x =
at + c 1
x = – at + c 2
⇒ Hyperbolic
and the exact solution is
f ( x + at ) + f ( x – at ) 1
u ( x, t ) = ----------------------------------------------- + -----2a
2
t
left
running
-1/a
1
( x + at )
∫(x – at ) g ( ξ )dξ
right
running
1/a
3
1
1
1
2
x0 - at0
x
x0 + at0
(a) Domain (zone) of Silence (region 1), dependence (region 2), and
influence (region 3)
(b) Propagation of discontinuity,
i.e., if I. C. is not a continuous fn, then ............
∆t
∆x
(c) CFL (Courant-Friedrichs-Lewy) Condition ==> a ------ ≤ 1
Page 2-5
Consider
Theoretical (actual) zone of dependence
Numerical zone of dependence
∆x
∆x
∆t
∆t
i.e.,
≤
Slope of
= 1/a
Slope of
= ∆t ⁄ ∆x
⇒
Page 2-6
∆t 1
------ ≤ --∆x a
⇒
∆t
a ------ ≤ 1
∆x
⇒
∆x
∆t ≤ -----
a
(d) Characteristic:
Defined as families of surfaces (curves) in a general 3D (2D),
unsteady flow, along which certain properties remain
constant or certain derivatives can become discontinuous.
e.g. 1, Linear Bergers equation
e.g. 2, Linear second-order wave equation
( 1 – M 2 )Φ xx + Φ yy = 0
if
= – 4 (1 – M2)
M < 1 ⇒ Subsonic Flow ⇒ Elliptic
M=1 ⇒ Sonic Flow ⇒ Parabolic
if
if
D
M > 1 ⇒ Supersonic Flow ⇒ Hyperbolic
Mach
Cone
Zone
of
Silence
Zone
of
Silence
Zone of
Action(denpendence)
Page 2-7
2-3 Model Equations
(1) Laplace’s/Poisson’s Equation:
 0
2
2

∂ Φ ∂ Φ
+
= 
∂x2 ∂y2

 f ( x, y )
(1-5)
Study Diffusion and Source
(2) Unsteady Heat Conduction Equation:
2
2
∂ T ∂ T
∂T
+
= α

2
2
∂t
∂y 
∂x
(1-6)
Study Unsteady and Diffusion
(3) 1st-Order Linear Wave Equation (Linear Bergers Eq.):
∂u
∂u
+a
= 0
∂t
∂x
(1-7)
Study the propagation of a wave moving to the right at a constant
speed “a”.
Page 2-8
(4) 1st-Order Nonlinear Wave Equation (Inviscid nonlinear
Bergers Eq.):
∂u
∂u
+u
= 0
∂t
∂x
(1-8)
Study the propagation of a nonlinear wave moving to the right at a
with speed “u” which is also the dependent variable.
(5) Viscous Burgers Equation:
2
∂u
∂ u
∂u
= ν
+u
∂t
∂x
∂x2
(1-9)
(a) Study Unsteady + Convection (nonlinear) + Diffusion
(b) Used to model momentum equation (e.g. x-momentum eq.)
(c) This eq. is the most often used to model the fluid flow and heat
transfer problems for numerical experiments.
(d) Stoke’s 1st Problem:
2
∂u
∂ u
= ν
∂t
∂x2
(e) if linear ==> 1D incompressible flow energy equation
Page 2-9
(1-10)
2
∂ T
∂T
∂T
= ν
+u
∂t
∂x
∂x2
(1-11)
(6) 2nd-Order Wave Equation:
2
2
∂ u
∂ u
= a2
∂ t2
∂x2
(1-12)
2-4 System of PDEs
In general, any higher order PDE can be converted to a
system of 1st-order PDEs
(1) 1st-Order PDEs
(a) Consider Eq. (1-12)
2
let
∂u
v =
∂t
∂u
w = a
∂x
 ∂v
∂ u

=
 ∂t
∂ t2
⇒
2

∂
u
∂v

=
∂x
∂ t ∂x
Page 2-10
2
∂ u
∂w
= a
∂x
∂x2
(A )
2
∂w
∂ u
= a
∂t
∂ x ∂t
(B)
∂v
∂w
= a
∂t
∂x
From (A), Eq. (1-12) Becomes
From (B), we have
∂w
∂v
= a
∂t
∂x
i.e.,
∂  v
∂ v
+ 0 –a   = 0
∂ t  w
– a 0 ∂ x  w
or
∂U
∂U
+A
= 0
∂t
∂x
where
v
U =  
 w
A =
0 –a
–a 0
The eigen values of A are
det A – λI = 0
⇒
λ –a
–a λ
=0
⇒ λ2 – a2 = 0
==> Two distinct real values ==> Hyperbolic!
Page 2-11
⇒ λ = ±a
(b) Consider the Laplace’s Eq.
2
2
∂ Φ ∂ Φ
+
= 0
2
2
∂x
∂y
let
u =
∂Φ
∂y
v =
∂Φ
∂x
⇒
∂  u
∂ u
+ 0 1   =0
∂ x  v
1 0 ∂ x  v
⇒ λ = ±i
==> Two distinct complex values (no real values) ==> Elliptic!
(c) Linear Unsteady System of Equations
∂φ
∂φ
∂φ
+A +B +ψ = 0
∂t
∂x
∂y
(1-13)
where matrices A and B are fns of t, x, and y
φ is the dependent variables and is a column vector
ψ is a fn of φ , x, and y, and is a column vector
Eq. (1-13) is
hyperbolic at a pt (x,t) if the eigen values of A are all real and distinct.
hyperbolic at a pt (y,t) if the eigen values of B are all real and distinct.
parabolic at a pt (x,t) if the eigen values of A are all real but < no. of eq.
parabolic at a pt (y,t) if the eigen values of B are all real but < no. of eq.
Page 2-12
Elliptic
at a pt (x,t) if the eigen values of A are complex.
Elliptic
at a pt (y,t) if the eigen values of B are complex.
Mixed H/E at a pt (x,t) if the eigen values of A are mixed real & complex.
Mixed H/E at a pt (y,t) if the eigen values of B are mixed real & complex.
(d) Linear Steady System of Equations
A
Define
∂φ
∂φ
+B +ψ = 0
∂x
∂y
(1-14)
2
H = R – 4PQ
where
P = det A
Q = det B
and
R =
a a
a1 a4
+ 3 2
b1 b4
b3 b2
Then if
H > 0 ==> Eq. (1-14) is hyperbolic
H = 0 ==> Eq. (1-14) is parabolic
H < 0 ==> Eq. (1-14) is
elliptic
Or rewrite (1-14) as
∂φ ∂φ
( Aî + Bĵ ) •  î + ĵ + ψ = 0
∂x ∂y
Page 2-13
(1-15)
and consider
n = n x î + n y ĵ
characteristic
In the characteristic direction ( n )
T = ( Aî + Bĵ ) • ( n x î + n y ĵ ) = An x + Bn y
A wave-like solution for the system may be obtained if
T = 0
An x + Bn y = 0
or
gives
 n x
 n x 2
Q  ----- + R  ----- + P = 0
 n y
 n y
H > 0 ⇒ Hy
 n x
–R± H
 ----- = ----------------------2Q
 n y
⇒ if
Page 2-14
H = 0 ⇒ Pa
H < 0 ⇒ Elli
Ex. 1: Steady, inviscid, and incompressible
∂u ∂v
+
= 0
∂x ∂y
u
∂u
∂u ∂p
+v +
= 0
∂x
∂y ∂x
u
∂v
∂v ∂p
+v +
= 0
∂x
∂y ∂y
In vector form
∂U
∂U
A
+B
= 0
∂x
∂y
0 1 0
B = v 0 0
0 v 1
1 0 0
A = u 0 1
0 u 0
2
2
==>
T = – ( un x + vn y ) [ ( n x ) + ( n y ) ] = 0
==>
ny
u
----- = – --------v
nx
and
ny
----- = ± – 1
nx
==> Mixed Hyperbolic/Elliptic System
Page 2-15
u
U = v
p
Ex. 2: Unsteady, 1D, inviscid, and Compressible ==> Euler’s Eq.
∂ρ
∂ρ
∂u
+u +ρ
= 0
∂t
∂x
∂x
∂u
∂u 1 ∂p
+ u + --= 0
∂t
∂x ρ ∂x
Ideal gas
a ≡ Sound Speed
∂u
∂p
∂p
+ u + ρa 2
= 0
∂x
∂t
∂x
In vector form
∂U
∂U
+A
= 0
∂t
∂x
u
ρ
A = 0
u
0
1
--ρ
0 ρa 2 u
ρ
U = u
p
Eigen value of A
u–λ
ρ
0
u–λ
0
0
1
--ρ
= 0
⇒
λ = u, u-a, u+a
⇒
Real and Distinct
⇒
ρa 2 u – λ
Page 2-16
Hyperbolic
(2) 2nd-Order PDEs
2nd-order ===> 1st-order ===> same as in (1)
Ex.: 2D, Steady, viscous, and incompressible
∂u ∂v
+
= 0
∂x ∂y
2
2
∂u
∂u
∂p 1  ∂ u ∂ u 
+
u +v
= –
+ ------ 

∂x
∂y
∂ x Re  ∂ x 2 ∂ y 2 
2
∂v

a = ∂x


∂v ⇒ ∂u = – ∂v = – b
b =
∂x
∂y
∂y


continuity
 c = ∂u
equation

∂y
set
2
∂v
∂v
∂p 1  ∂ v ∂ v 
+
u +v
= –
+ ------ 

∂x
∂y
∂ y Re  ∂ x 2 ∂ y 2 
==> In vector form
n
A
∂U
∂U
+B
= C
∂x
∂y
0
1
0
A = 0
0
0
0
0
0
0
0
0
0 0
0
0
0
1
0
1
-----Re
0
0
0
1
0
0
0
0
0 1
1
0 0 – ------ 0 0 0
Re
1
0
0
B = 0
0
1
0
0
0
0
–1
0
0
0
0
1
0 0 0
0
1
0 0 0 – -----Re
Page 2-17
0
0
0
0
1
– -----Re
0
0
0
0
0
1
0
u
v
U = a
b
c
p
⇒T=
0
0
0
0
n n
0
x y
0 0 –n
0
0
0
n
0
0
y
0
0 0
0
0 0
0
y
n
x
0 0 – -----Re
n
x
0
x
n
n
y
x
------ – ------ n
Re Re x
n
y
– ------ 0 n
y
Re
y
n
==>
2
2
2 2
1
T = ------ ( n y ) [ ( n x ) + ( n y ) ] = 0
Re
2
2
( nx ) + ( ny ) = 0
==>
,
ny = 0
 n y 2
 ----- + 1 = 0
 n x
or
==>
Imaginary
==>
Mixed Elliptic/Hyperbolic System
2-5 Initial and Boundary Conditions
I.C. :
at
t = 0
φ = φ ( r ) = φ ( x, y, z )
B.C. : 1st kind (Dorochlet): e.g., no slip, prescribed T
2nd kind (Neumann): e.g., adiabatic wall
3rd kind (Robin): e.g., convective B.C.
Mixed kind
2nd
3rd
1st
Page 2-18
2-6 Remarks
(1) Grid: even with
better if ∇φ ≈ C
(2) PDEs ==> Algebraic ==> (a) Consistency, (b) Stability, (c) Convergency
(3) The Conservation Form of G.E.s
e.g.1: Continuity Equation
∂ρ ∂
∂
∂
+ ( ρu ) + ( ρv ) + ( ρw ) = 0
∂t ∂x
∂y
∂z
or
∂ρ
∂ρ
∂u
∂ρ
∂v
∂ρ
∂w
+u +ρ +v +ρ +w +ρ
= 0
∂t
∂x
∂x
∂y
∂y
∂z
∂z
==> the coefficients of the derivative terms are either constant
or, if variable, their derivatives appear nowhere in the eq.
e.q.2: Unsteady Heat Conduction equation
∂T
∂  ∂T
ρc
=
κ
∂t
∂ x  ∂ x
or
2
∂T
∂ T  ∂κ  ∂T
ρc
= κ
+
 ∂ x  ∂ x 
∂t
2
∂x
Page 2-19