# hyperbolic pdes numerical methods for pdes spring 2007 jim e. jones

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

• PDE classified by discriminant: b2-4ac.Negative discriminant = Elliptic PDE. Example Laplaces equation

Zero discriminant = Parabolic PDE. Example Heat equation

Positive discriminant = Hyperbolic PDE. Example Wave equation Partial Differential Equations (PDEs) :2nd order model problems

• Example: Hyperbolic Equation (Infinite Domain)Wave equationInitial Conditions

• Example: Hyperbolic Equation (Infinite Domain)Wave equationInitial ConditionsSolution (verify)

• Hyperbolic Equation: characteristic curvesx-ct=constantx+ct=constantxt(x,t)

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

• Example: Hyperbolic Equation (Infinite Domain)x-ct=constantx+ct=constantxt(x,t)The region bounded by the characteristics is called the domain of dependence of the PDE.

• Example: Hyperbolic Equation (Infinite Domain)Wave equationInitial Conditions

• Example: Hyperbolic Equation (Infinite Domain)t=.01t=.1t=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 equationWave equationChoose step size h in space and k in timehktx

• Finite difference method for wave equationWave equationChoose step size h in space and k in time

• Finite difference method for wave equationWave equationChoose step size h in space and k in time

Solve for ui,j+1

• Finite difference method for wave equationStencil involves u values at 3 different time levelshktx

• Finite difference method for wave equationCant use this for first time step.hktxU at initial time given by initial condition.ui,0 = f(xi)

• Finite difference method for wave equationUse initial derivative to make first time step.hktxU at initial time given by initial condition

• Finite difference method for wave equationWhich discrete values influence ui,j+1 ?hktx

• Finite difference method for wave equationWhich discrete values influence ui,j+1 ?hktx

• Finite difference method for wave equationWhich discrete values influence ui,j+1 ?hktx

• Finite difference method for wave equationWhich discrete values influence ui,j+1 ?hktx

• Finite difference method for wave equationWhich discrete values influence ui,j+1 ?hktx

• Domain of dependence for finite difference methodThose discrete values influence ui,j+1 define the discrete domain of dependencehktx

• 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 hktxx-ct=constantx+ct=constant

• CFL (Courant, Friedrichs, Lewy) Condition Possibly stable: domain of dependence of PDE is inside discrete domain of dependence hktxx-ct=constantx+ct=constant

• CFL (Courant, Friedrichs, Lewy) Condition Boundary of unstable: domain of dependence of PDE is discrete domain of dependence hktxx-ct=constantx+ct=constant

• CFL (Courant, Friedrichs, Lewy) Condition Boundary of unstable: domain of dependence of PDE is discrete domain of dependence hktxx-ct=constantx+ct=constantk/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 equationInitial Conditions

• Hyperbolic Equation: characteristic curves on finite domainx-ct=constantx+ct=constantxt(x,t)x=bx=a

• Hyperbolic Equation: characteristic curves on finite domainx-ct=constantx+ct=constantxt(x,t)x=bx=aValue is influenced by boundary values. Represents incoming waves

• Example: Hyperbolic Equation (Finite Domain)Wave equationInitial Conditions

Boundary Conditions

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