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Hyperbolic PDEs Numerical Methods for PDEs Spring 2007

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Hyperbolic PDEs Numerical Methods for PDEs Spring 2007. Jim E. Jones. Partial Differential Equations (PDEs) : 2 nd order model problems. PDE classified by discriminant: b 2 -4ac. Negative discriminant = Elliptic PDE. Example Laplace’s equation - PowerPoint PPT Presentation
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Hyperbolic PDEs Numerical Methods for PDEs Spring 2007 Jim E. Jones
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Page 1: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Hyperbolic PDEs

Numerical Methods for PDEs Spring 2007

Jim E. Jones

Page 2: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

• 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

Page 3: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Example: Hyperbolic Equation (Infinite Domain)

02 xxtt ucu

)()0,(

)()0,(

xgxu

xfxu

t

Wave equation

Initial Conditions

),0(),(),( tx

Page 4: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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)

Page 5: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Hyperbolic Equation: characteristic curves

x-ct=constantx+ct=constant

x

t

(x,t)

Page 6: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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),(

Page 7: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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.

Page 8: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Example: Hyperbolic Equation (Infinite Domain)

0 xxtt uu

0)0,(

)exp()0,( 2

xu

xxu

t

Wave equation

Initial Conditions

),0(),(),( tx

Page 9: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Example: Hyperbolic Equation (Infinite Domain)

t=.01 t=.1

t=1 t=10

Page 10: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

• 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

Page 11: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Finite difference method for wave equation

02 xxtt ucuWave equation

Choose step size h in space and k in time

h

k

t x

Page 12: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 13: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 14: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 15: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 16: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 17: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 18: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 19: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 20: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 21: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 22: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 23: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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.

Page 24: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 25: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 26: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 27: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 28: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

/

/

Page 29: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

/

Page 30: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Example: Hyperbolic Equation (Finite Domain)

02 xxtt ucu

)()0,(

)()0,(

xgxu

xfxu

t

Wave equation

Initial Conditions

),0(),(),( Tbatx

Page 31: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Hyperbolic Equation: characteristic curves on finite domain

x-ct=constantx+ct=constant

x

t

(x,t)

x=bx=a

Page 32: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

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

Page 33: Hyperbolic PDEs                   Numerical Methods for PDEs Spring 2007

Example: Hyperbolic Equation (Finite Domain)

02 xxtt ucu

)()0,(

)()0,(

xgxu

xfxu

t

Wave equation

Initial Conditions

Boundary Conditions

),0(),(),( Tbatx

)(),(

)(),(

ttbu

ttau


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