Hyperbolic PDEs Numerical Methods for PDEs Spring 2007 Jim E. Jones

PDE classified by discriminant: b 2 -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) : 2 nd order model problems

Example: Hyperbolic Equation (Infinite Domain) Wave equation Initial Conditions

Example: Hyperbolic Equation (Infinite Domain) Wave equation Initial Conditions Solution (verify)

Hyperbolic Equation: characteristic curves x-ct=constant x+ct=constant x t (x,t)

Example: Hyperbolic Equation (Infinite Domain) x-ct=constant x+ct=constant x t (x,t) The point (x,t) is influenced only by initial conditions bounded by characteristic curves.

Example: Hyperbolic Equation (Infinite Domain) x-ct=constant x+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) Wave equation Initial Conditions

Example: Hyperbolic Equation (Infinite Domain) t=.01t=.1 t=1t=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 Wave equation Choose step size h in space and k in time h k t x

Finite difference method for wave equation Wave equation Choose step size h in space and k in time

Finite difference method for wave equation Wave equation Choose step size h in space and k in time Solve for u i,j+1

Finite difference method for wave equation Stencil involves u values at 3 different time levels h k t x

Finite difference method for wave equation Can’t use this for first time step. h k t x U at initial time given by initial condition. u i,0 = f(x i )

Finite difference method for wave equation Use initial derivative to make first time step. h k t x U at initial time given by initial condition

Finite difference method for wave equation Which discrete values influence u i,j+1 ? h k t x

Finite difference method for wave equation Which discrete values influence u i,j+1 ? h k t x

Finite difference method for wave equation Which discrete values influence u i,j+1 ? h k t x

Finite difference method for wave equation Which discrete values influence u i,j+1 ? h k t x

Finite difference method for wave equation Which discrete values influence u i,j+1 ? h k t x

Domain of dependence for finite difference method Those discrete values influence u i,j+1 define the discrete domain of dependence h k t x

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=constant x+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=constant x+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=constant x+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=constant x+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.

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.

Example: Hyperbolic Equation (Finite Domain) Wave equation Initial Conditions

Hyperbolic Equation: characteristic curves on finite domain x-ct=constant x+ct=constant x t (x,t) x=b x=a

Hyperbolic Equation: characteristic curves on finite domain x-ct=constant x+ct=constant x t (x,t) x=b x=a Value is influenced by boundary values. Represents incoming waves

Example: Hyperbolic Equation (Finite Domain) Wave equation Initial Conditions Boundary Conditions