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

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