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