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Hyperbolic PDEs Numerical Methods for PDEs Spring 2007 Jim E. Jones
Hyperbolic PDEs
Numerical Methods for PDEs Spring 2007
Jim E. Jones
• PDE classified by discriminant: b2-4ac.– Negative discriminant = Elliptic PDE. Example
Laplace’s equation
– Zero discriminant = Parabolic PDE. Example Heat equation
– Positive discriminant = Hyperbolic PDE. Example Wave equation
Partial Differential Equations (PDEs) :2nd order model problems
0 yyxx uu
0 xxt uu
0 ttxx uu
gfueuducubuau yxyyxyxx
Example: Hyperbolic Equation (Infinite Domain)
02 xxtt ucu
)()0,(
)()0,(
xgxu
xfxu
t
Wave equation
Initial Conditions
),0(),(),( tx
Example: Hyperbolic Equation (Infinite Domain)
02 xxtt ucu
)()0,(
)()0,(
xgxu
xfxu
t
),0(),(),( txWave equation
Initial Conditions
ctx
ctx
dyygc
ctxfctxftxu )(2
1)]()([
2
1),(
Solution (verify)
Hyperbolic Equation: characteristic curves
x-ct=constantx+ct=constant
x
t
(x,t)
Example: Hyperbolic Equation (Infinite Domain)
x-ct=constantx+ct=constant
x
t
(x,t) The point (x,t) is influenced only by initial conditions bounded by characteristic curves.
ctx
ctx
dyygc
ctxfctxftxu )(2
1)]()([
2
1),(
Example: Hyperbolic Equation (Infinite Domain)
x-ct=constantx+ct=constant
x
t
(x,t) The region bounded by the characteristics is called the domain of dependence of the PDE.
Example: Hyperbolic Equation (Infinite Domain)
0 xxtt uu
0)0,(
)exp()0,( 2
xu
xxu
t
Wave equation
Initial Conditions
),0(),(),( tx
Example: Hyperbolic Equation (Infinite Domain)
t=.01 t=.1
t=1 t=10
• Typically describe time evolution with no steady state.– Model problem: Describe the time evolution of the
wave produced by plucking a string.
• Initial conditions have only local effect – The constant c determines the speed of wave
propagation.
Hyperbolic PDES
Finite difference method for wave equation
02 xxtt ucuWave equation
Choose step size h in space and k in time
h
k
t x
Finite difference method for wave equation
02 xxtt ucu
)2(1
)),(),(2),((1
),(
)2(1
)),(),(2),((1
),(
,1,,12
2
1,,1,2
2
jijiji
jijijijixx
jijiji
jijijijitt
uuuh
thxutxuthxuh
txu
uuuk
ktxutxuktxuk
txu
Wave equation
Choose step size h in space and k in time
Finite difference method for wave equation
02 xxtt ucu
0)2()2(1
,1,,12
2
1,,1,2 jijijijijiji uuu
h
cuuu
k
Wave equation
Choose step size h in space and k in time
Solve for ui,j+1
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Stencil involves u values at 3 different time levels
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Can’t use this for first time step.
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
U at initial time given by initial condition.ui,0 = f(xi)
)()0,(
)()0,(
xgxu
xfxu
t
Finite difference method for wave equation
Use initial derivative to make first time step.
h
k
t x
iii
iii
fkgu
xguuk
1,
0,1, )(1
U at initial time given by initial condition
)()0,(
)()0,(
xgxu
xfxu
t
Finite difference method for wave equation
Which discrete values influence ui,j+1 ?
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Which discrete values influence ui,j+1 ?
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Which discrete values influence ui,j+1 ?
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Which discrete values influence ui,j+1 ?
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Finite difference method for wave equation
Which discrete values influence ui,j+1 ?
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
Domain of dependence for finite difference method
Those discrete values influence ui,j+1 define the discrete domain of dependence
h
k
t x
)2(2 ,1,,12
22
1,,1, jijijijijiji uuuh
kcuuu
CFL (Courant, Friedrichs, Lewy) Condition
A necessary condition for an explicit finite difference scheme for a hyperbolic PDE to be stable is that for each mesh point the domain of dependence of the PDE must lie within the discrete domain of dependence.
CFL (Courant, Friedrichs, Lewy) Condition
Unstable: part of domain of dependence of PDE is outside discrete domain of dependence
h
k
t x
x-ct=constantx+ct=constant
CFL (Courant, Friedrichs, Lewy) Condition
Possibly stable: domain of dependence of PDE is inside discrete domain of dependence
h
k
t x
x-ct=constantx+ct=constant
CFL (Courant, Friedrichs, Lewy) Condition
Boundary of unstable: domain of dependence of PDE is discrete domain of dependence
h
k
t x
x-ct=constantx+ct=constant
CFL (Courant, Friedrichs, Lewy) Condition
Boundary of unstable: domain of dependence of PDE is discrete domain of dependence
h
k
t x
x-ct=constantx+ct=constant
k/h=1/c
CFL (Courant, Friedrichs, Lewy) Condition
A necessary condition for an explicit finite difference scheme for a hyperbolic PDE to be stable is that for each mesh point the domain of dependence of the PDE must lie within the discrete domain of dependence.
cxt
chk
/
/
CFL (Courant, Friedrichs, Lewy) Condition
The constant c is the wave speed, CFL condition says that a wave cannot cross more than one grid cell in one time step.
xtc
cxt
/
Example: Hyperbolic Equation (Finite Domain)
02 xxtt ucu
)()0,(
)()0,(
xgxu
xfxu
t
Wave equation
Initial Conditions
),0(),(),( Tbatx
Hyperbolic Equation: characteristic curves on finite domain
x-ct=constantx+ct=constant
x
t
(x,t)
x=bx=a
Hyperbolic Equation: characteristic curves on finite domain
x-ct=constantx+ct=constant
x
t
(x,t)
x=bx=a
Value is influenced by boundary values. Represents incoming waves
Example: Hyperbolic Equation (Finite Domain)
02 xxtt ucu
)()0,(
)()0,(
xgxu
xfxu
t
Wave equation
Initial Conditions
Boundary Conditions
),0(),(),( Tbatx
)(),(
)(),(
ttbu
ttau
![Numerical Analysis of the Shallow Water Equations · Moreover, in the field of hyperbolic PDEs finite difference method is still most widely used. We note that PRAAGMAN [35] has implemented](https://static.documents.pub/doc/80x56/60620fae389c63538a63c662/numerical-analysis-of-the-shallow-water-equations-moreover-in-the-field-of-hyperbolic.jpg)
