# Hyperbolic PDEs Numerical Methods for PDEs Spring 2007

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"Hyperbolic PDEs Numerical Methods for PDEs Spring 2007"

```					    Hyperbolic PDEs
Numerical Methods for PDEs
Spring 2007
Jim E. Jones
Partial Differential Equations (PDEs) :
2nd order model problems
au xx  bu xy  cu yy  du x  eu y  fu  g
• PDE classified by discriminant: b2-4ac.
– Negative discriminant = Elliptic PDE. Example
Laplace’s equation
u xx  u yy  0

– Zero discriminant = Parabolic PDE. Example Heat
equation
ut  u xx  0

– Positive discriminant = Hyperbolic PDE. Example
Wave equation
u xx  utt  0
Example: Hyperbolic Equation (Infinite Domain)

Wave equation

utt  c 2u xx  0        ( x, t )  (, )  (0, )

Initial Conditions

u ( x ,0 )  f ( x )
u t ( x ,0 )  g ( x )
Example: Hyperbolic Equation (Infinite Domain)

Wave equation
utt  c 2u xx  0                  ( x, t )  (, )  (0, )

Initial Conditions
u ( x ,0 )  f ( x )
u t ( x ,0 )  g ( x )

Solution (verify)
x  ct
1                                 1
u ( x, t )  [ f ( x  ct )  f ( x  ct )] 
2                                       g ( y)dy
2c x ct
Hyperbolic Equation: characteristic curves

x+ct=constant      x-ct=constant
t

(x,t)

x
Example: Hyperbolic Equation (Infinite Domain)

x+ct=constant      x-ct=constant
x  ct
t                                       1                                 1
u ( x, t )  [ f ( x  ct )  f ( x  ct )] 
2                                      ctg ( y)dy
2c x 

(x,t)         The point (x,t) is
influenced only by initial
conditions bounded by
characteristic curves.

x
Example: Hyperbolic Equation (Infinite Domain)

x+ct=constant      x-ct=constant
t

(x,t)    The region bounded by
the characteristics is
called the domain of
dependence of the PDE.

x
Example: Hyperbolic Equation (Infinite Domain)

Wave equation
utt  u xx  0          ( x, t )  (, )  (0, )

Initial Conditions

u( x,0)  exp( x 2 )
ut ( x,0)  0
Example: Hyperbolic Equation (Infinite Domain)

t=.01                        t=.1

t=1                         t=10
Hyperbolic PDES

• 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.
Finite difference method for wave equation

Wave equation       utt  c 2u xx  0

Choose step size h in space and k in time

k

t    x                          h
Finite difference method for wave equation

Wave equation

utt  c 2u xx  0

Choose step size h in space and k in time
1
utt ( xi , t j )   2
(u ( xi , t j  k )  2u ( xi , t j )  u ( xi , t j  k ))
k
1
 2 (ui , j 1  2ui , j  ui , j 1 )
k
1
u xx ( xi , t j )  2 (u ( xi  h, t j )  2u ( xi , t j )  u ( xi  h, t j ))
h
1
 2 (ui 1, j  2ui , j  ui 1, j )
h
Finite difference method for wave equation

Wave equation                 utt  c 2u xx  0

Choose step size h in space and k in time
1                                       c2
2
(ui , j 1  2ui , j  ui , j 1 )  2 (ui 1, j  2ui , j  ui 1, j )  0
k                                       h
Solve for ui,j+1
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Stencil involves u values at 3 different time levels

k

t       x                                h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Can’t use this for first time step.

U at initial time given
by initial condition.
ui,0 = f(xi)

k
u ( x ,0 )  f ( x )
t                                                    u t ( x ,0 )  g ( x )
x                                h
Finite difference method for wave equation
Use initial derivative to make first time step.
1
ui,1  ui,0   g ( xi )
k
ui ,1  kgi  f i

U at initial time given
by initial condition

k
u ( x ,0 )  f ( x )
t                                             u t ( x ,0 )  g ( x )
x                              h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Which discrete values influence ui,j+1 ?

k

t       x                                h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Which discrete values influence ui,j+1 ?

k

t       x                                h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Which discrete values influence ui,j+1 ?

k

t       x                                h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Which discrete values influence ui,j+1 ?

k

t       x                                h
Finite difference method for wave equation
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Which discrete values influence ui,j+1 ?

k

t       x                                h
Domain of dependence for finite difference
method
c2k 2
ui , j 1  2ui , j  ui , j 1  2 (ui 1, j  2ui , j  ui 1, j )
h
Those discrete values influence ui,j+1 define the discrete
domain of dependence

k

t      x                               h
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

x+ct=constant              x-ct=constant

k

t    x                    h
CFL (Courant, Friedrichs, Lewy) Condition

Possibly stable: domain of dependence of PDE is inside
discrete domain of dependence

x+ct=constant            x-ct=constant

k

t    x                    h
CFL (Courant, Friedrichs, Lewy) Condition

Boundary of unstable: domain of dependence of PDE is
discrete domain of dependence

x+ct=constant            x-ct=constant

k

t    x                    h
CFL (Courant, Friedrichs, Lewy) Condition

Boundary of unstable: domain of dependence of PDE is
discrete domain of dependence

x+ct=constant            x-ct=constant

k/h=1/c

k

t    x                    h
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.

k  h/c
t   x / c
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.

t   x / c
ct  x
Example: Hyperbolic Equation (Finite Domain)

Wave equation

utt  c 2u xx  0          ( x, t )  (a, b)  (0, T )

Initial Conditions

u ( x ,0 )  f ( x )
u t ( x ,0 )  g ( x )
Hyperbolic Equation: characteristic curves
on finite domain

x+ct=constant      x-ct=constant
t

(x,t)

x

x=a                x=b
Hyperbolic Equation: characteristic curves
on finite domain

x+ct=constant      x-ct=constant
t

(x,t)   Value is influenced by boundary
values. Represents incoming
waves

x

x=a                x=b
Example: Hyperbolic Equation (Finite Domain)

Wave equation

utt  c 2u xx  0          ( x, t )  (a, b)  (0, T )

Initial Conditions

u ( x ,0 )  f ( x )
u t ( x ,0 )  g ( x )

Boundary Conditions
u (a, t )   (t )
u (b, t )   (t )

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