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Hyperbolic PDEs Numerical Methods for PDEs Spring 2007

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Hyperbolic PDEs Numerical Methods for PDEsSpring 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

Wave equation

Initial Conditions

Wave equation

Initial Conditions

Solution (verify)

x+ct=constant

x-ct=constant

t

(x,t)

x

x+ct=constant

x-ct=constant

t

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

(x,t)

x

x+ct=constant

x-ct=constant

t

The region bounded by the characteristics is called the domain of dependence of the PDE.

(x,t)

x

Wave equation

Initial Conditions

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.

Wave equation

Choose step size h in space and k in time

k

t

x

h

Wave equation

Choose step size h in space and k in time

Wave equation

Choose step size h in space and k in time

Solve for ui,j+1

Stencil involves u values at 3 different time levels

k

t

x

h

Can’t use this for first time step.

U at initial time given by initial condition.

ui,0 = f(xi)

k

t

x

h

Use initial derivative to make first time step.

U at initial time given by initial condition

k

t

x

h

Which discrete values influence ui,j+1 ?

k

t

x

h

Which discrete values influence ui,j+1 ?

k

t

x

h

Which discrete values influence ui,j+1 ?

k

t

x

h

Which discrete values influence ui,j+1 ?

k

t

x

h

Which discrete values influence ui,j+1 ?

k

t

x

h

Those discrete values influence ui,j+1 define the discrete domain of dependence

k

t

x

h

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.

Unstable: part of domain of dependence of PDE is outside discrete domain of dependence

x-ct=constant

x+ct=constant

k

t

x

h

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

x+ct=constant

x-ct=constant

k

t

x

h

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

x+ct=constant

x-ct=constant

k

t

x

h

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

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.

The constant c is the wave speed, CFL condition says that a wave cannot cross more than one grid cell in one time step.

Wave equation

Initial Conditions

x+ct=constant

x-ct=constant

t

(x,t)

x

x=a

x=b

x+ct=constant

x-ct=constant

t

Value is influenced by boundary values. Represents incoming waves

(x,t)

x

x=a

x=b

Wave equation

Initial Conditions

Boundary Conditions