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Computational Fluid Dynamics IPPPPIIIIWWWW
Theory ofPartial Differential
Equations
Instructor: Hong G. ImUniversity of Michigan
Fall 2001
Computational Fluid Dynamics IPPPPIIIIWWWW
• Basic Properties of PDE
• Quasi-linear First Order Equations
- Characteristics
- Linear and Nonlinear Advection Equations
• Quasi-linear Second Order Equations
- Classification: hyperbolic, parabolic, elliptic
- Domain of Dependence/Influence
- Ill-posed Problems
Outline
Computational Fluid Dynamics IPPPPIIIIWWWW• The order of PDE is determined by the highest
derivatives• Linear if no powers or products of the unknown
functions or its partial derivatives are present.
• Quasi-linear if it is true for the partial derivatives of highest order.
Definitions
02, =+∂∂+
∂∂=
∂∂+
∂∂ xf
yf
xff
yfy
xfx
222
2
2
2
, fyfy
xfxf
yf
xff =
∂∂+
∂∂=
∂∂+
∂∂
Computational Fluid Dynamics IPPPPIIIIWWWW• The general solution of PDE are arbitrary functions
of specific functions.
Definitions
)2(;02 yxfyf
xf +==
∂∂−
∂∂ ψ
)()(
;02
22
2
2
dycxbyaxfyfC
yxfB
xfA
+++=
=∂∂+
∂∂∂+
∂∂
φψ
Computational Fluid Dynamics IPPPPIIIIWWWW
Quasi-linear first order partial differential equations
Computational Fluid Dynamics IPPPPIIIIWWWW
cyfb
xfa =
∂∂+
∂∂
),,(),,(),,(
fyxccfyxbbfyxaa
===
Consider the quasi-linear first order equation
where the coefficients are functions of x,y, and f, but not the derivatives of f:
Computational Fluid Dynamics IPPPPIIIIWWWW
The solution of this equations defines a single valued surface f(x,y) in three-dimensional space:
f=f(x,y)
f
x
y
Computational Fluid Dynamics IPPPPIIIIWWWW
dyyfdx
xfdf
∂∂+
∂∂=
An arbitrary change in f is given by:
f=f(x,y)
f
x
y
f
dxdy
df
Computational Fluid Dynamics IPPPPIIIIWWWW
cyfb
xfa =
∂∂+
∂∂
dyyfdx
xfdf
∂∂+
∂∂=
0),,(1,, =⋅
−
∂∂
∂∂ cba
yf
xf
0),,(1,, =⋅
−
∂∂
∂∂ dfdydx
yf
xf
⇒
⇒
The original equation and the conditions for a small change can be rewritten as:
Computational Fluid Dynamics IPPPPIIIIWWWW
0),,(1,, =⋅
−
∂∂
∂∂ cba
yf
xf 0),,(1,, =⋅
−
∂∂
∂∂ dfdydx
yf
xf
),,(),,( cbadsdfdydx =
;;; cdsdfb
dsdya
dsdx ===
ba
dydx =
⇒
⇒
Both
Picking the displacement in the direction of (a,b,c)
Separating the components
Characteristics
Computational Fluid Dynamics IPPPPIIIIWWWW
;;; cdsdfb
dsdya
dsdx ===
ba
dydx =
The three equations
Given the initial conditions:
);();();( ooo sffsyysxx ===
the equations can be integrated in time
Computational Fluid Dynamics IPPPPIIIIWWWW
x(s); y(s); f (s)
x
y
x(0); y(0); f (0)
Computational Fluid Dynamics IPPPPIIIIWWWW
0=∂∂+
∂∂
xfU
tf
;0;;1 ===dsdfU
dsdx
dsdt
;0; == dfUdtdx
Consider the linear advection equation
The characteristics are given by:
or
Which shows that the solution moves along straight characteristics without changing its value
Computational Fluid Dynamics IPPPPIIIIWWWW
0=∂∂+
∂∂
xfU
tf
Graphically:
Notice that these results are specific for:
ft
f
x
Computational Fluid Dynamics IPPPPIIIIWWWW
fxfU
tf −=
∂∂+
∂∂
Add a source
t
f
f
x
;;;1 fdsdfU
dsdx
dsdt −===
tff
fdtdfU
dtdx
−=⇒
−==
e)0(
;
The characteristics are given by:
or
Moving wave with decaying amplitude
Computational Fluid Dynamics IPPPPIIIIWWWW
0=∂∂+
∂∂
xff
tf
;0;;1 ===dsdff
dsdx
dsdt
;0; == dffdtdx
Consider a nonlinear (quasi-linear) advection equation
The characteristics are given by:
or
The slope of the characteristics depends on the value of f(x,t).
Computational Fluid Dynamics IPPPPIIIIWWWW
t
fx
t3
t2
t1
Multi-valued solution
Computational Fluid Dynamics IPPPPIIIIWWWW
02
2
=∂∂−
∂∂+
∂∂
xf
xff
tf ε
Why unphysical solutions?
- Because mathematical equation neglects some physical process (dissipation)
Additional condition is required to pick out the physicallyRelevant solution
Correct solution is expected from Burgers equation with
Burgers Equation
0→εEntropy Condition (to be continued)
Computational Fluid Dynamics IPPPPIIIIWWWW
Quasi-linear second order partial differential equations
Computational Fluid Dynamics IPPPPIIIIWWWW
dy
fcyxfb
xfa =
∂∂+
∂∂∂+
∂∂
2
22
2
2
a = a(x,y, f )b = b(x, y, f )c = c(x, y, f )
d = d(x, y, f )
Consider a general quasi-linear PDE
where
Computational Fluid Dynamics IPPPPIIIIWWWW
dy
fcyxfb
xfa =
∂∂+
∂∂∂+
∂∂
2
22
2
2
xfv
∂∂=
yfw
∂∂=
dywc
yvb
xva =
∂∂+
∂∂+
∂∂
0=∂∂−
∂∂
yv
xw
∂∂
∂=∂∂
∂xyf
yxf 22
Starting with
Define
PDE becomes
with
Computational Fluid Dynamics IPPPPIIIIWWWW
=
∂∂∂∂
−
+
∂∂∂∂
001ad
ywyvacab
xwxv
sAuu yx =+
dywc
yvb
xva =
∂∂+
∂∂+
∂∂ 0=
∂∂−
∂∂
yv
xw
In matrix form
=
wv
u
Are there lines in the x-y plane, along which the solution is determined by an ordinary differential equation?
Computational Fluid Dynamics IPPPPIIIIWWWW
yv
xv
xy
yv
xv
dxdv
∂∂+
∂∂=
∂∂
∂∂+
∂∂= α
∂∂=xyα
adl
yv
xwl
yw
ac
yv
ab
xvl 121 =
∂∂−
∂∂+
∂∂+
∂∂+
∂∂
adl
yw
xwl
yv
xvl 121 =
∂∂+
∂∂+
∂∂+
∂∂ αα
Is this ever equal to
yw
xw
xy
yw
xw
dxdw
∂∂+
∂∂=
∂∂
∂∂+
∂∂= α
Computational Fluid Dynamics IPPPPIIIIWWWW
α21 lacl =
α121 llabl =−
adl
yv
xwl
yw
ac
yv
ab
xvl 121 =
∂∂−
∂∂+
∂∂+
∂∂+
∂∂
adl
yw
xwl
yv
xvl 121 =
∂∂+
∂∂+
∂∂+
∂∂ αα
Computational Fluid Dynamics IPPPPIIIIWWWWSolution determined by ODE (Characteristics exist) only if
α
α
21
121
lacl
llabl
=
=−
0;001
2
1 =−
=
−−−
αα
αjiji lAl
ll
acab
Or, in matrix form:
Compatibility Relation
Computational Fluid Dynamics IPPPPIIIIWWWW
0=+
−−
ac
ab αα
( )acbba
421 2 −±=α
=
−−−
001
2
1
ll
acab
αα
The determinant is:
Solving for α
0T =− IA α
Computational Fluid Dynamics IPPPPIIIIWWWW
042 =− acb
042 >− acb
042 <− acb
Two real characteristics
One real characteristics
No real characteristics
( )acbba
421 2 −±=α
Computational Fluid Dynamics IPPPPIIIIWWWW
02
22
2
2
=∂∂−
∂∂
yfc
xf
dy
fcyxfb
xfa =
∂∂+
∂∂∂+
∂∂
2
22
2
2
Examples
Comparing with the standard form
shows that ;0;;0;1 2 =−=== dccba
041404 2222 >=⋅⋅+=− ccacb
Hyperbolic
Wave Equation
Computational Fluid Dynamics IPPPPIIIIWWWW
02
2
2
2
=∂∂+
∂∂
yf
xf
Examples
Comparing with the standard form
shows that ;0;1;0;1 ==== dcba
0411404 22 <−=⋅⋅−=− acb
Elliptic
Laplace Equation
dy
fcyxfb
xfa =
∂∂+
∂∂∂+
∂∂
2
22
2
2
Computational Fluid Dynamics IPPPPIIIIWWWW
2
2
yfD
xf
∂∂=
∂∂
dy
fcyxfb
xfa =
∂∂+
∂∂∂+
∂∂
2
22
2
2
Examples
Comparing with the standard form
shows that ;0;;0;0 =−=== dDcba
00404 22 =⋅⋅+=− Dacb
Parabolic
Heat Equation
Computational Fluid Dynamics IPPPPIIIIWWWW
02
2
2
2
=∂∂+
∂∂
yf
xf EllipticLaplace equation
02
22
2
2
=∂∂−
∂∂
yfc
xf
HyperbolicWave equation
2
2
yfD
xf
∂∂=
∂∂ ParabolicHeat equation
Examples
Computational Fluid Dynamics IPPPPIIIIWWWW
( )acbba
421 2 −±=α
xy
∂∂=α
Hyperbolic Parabolic Elliptic
042 >− acb 042 =− acb 042 <− acb
Computational Fluid Dynamics IPPPPIIIIWWWW
02
22
2
2
=∂∂−
∂∂
xfc
tf
First write the equation as a system of first order equations
;;xfcv
tfu
∂∂=
∂∂=
0
0
=∂∂−
∂∂
=∂∂−
∂∂
xuc
tv
xvc
tuyielding
Introduce
Wave equation
from the PDE
sincextf
txf
∂∂∂=
∂∂∂ 22
Computational Fluid Dynamics IPPPPIIIIWWWW
Domain of Domain of InfluenceInfluence
Domain of Domain of DependenceDependence
t
x
P
Two characteristic linescdx
dtcdx
dt 1;1 −=+=
cdxdt 1+=
cdxdt 1−=
Computational Fluid Dynamics IPPPPIIIIWWWW
Ill-posed problems
Computational Fluid Dynamics IPPPPIIIIWWWWIll-posed Problems
2
2
2
2
xf
tf
∂∂−=
∂∂
Consider the initial value problem:
02
2
2
2
=∂∂+
∂∂
xf
tf
This is simply Laplace’s equation
which has a solution if ∂f/∂x or f are given on the boundaries
Computational Fluid Dynamics IPPPPIIIIWWWWIll-posed Problems
Here, however, this equation appeared as an initial value problem, where the only boundary conditions available are at t = 0. Since this is a second order equation we will need two conditions, we may assume are that f and ∂f/∂x are specified at t = 0.
ikxetaf )(=
akdt
ad 22
2
=
Consider separation of variables
PDE becomes
Computational Fluid Dynamics IPPPPIIIIWWWWIll-posed Problems
)0( ;)0( ikaa
akdt
ad 22
2
=
determined by initial conditions
General solutionktkt BeAeta −+=)(
BA,
Therefore,
∞→)(ta as ∞→t
Ill-posed Problem
Computational Fluid Dynamics IPPPPIIIIWWWWIll-posed Problems: Summary
IVP ill-posedBVP well-posed
IVP well-posed BVP ill-posed
IVP well posedBVP ill-posed
N/ANo characteristics
Infinite speed ofcharacteristics
Finite speeds of characteristics
Solutions are always smooth
Solutions are always smooth
Solutions may not be smooth
Data needed over entire boundary
Data needed over part of boundary
Data needed over part of boundary
No real characteristics
One real characteristics
Two real characteristics
EllipticParabolicHyperbolic
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