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

2. Partial Differential Equations (PDEs) : 2nd order model problems
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

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

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

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

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

7. Example: Hyperbolic Equation (Infinite Domain)
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

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

9. Example: Hyperbolic Equation (Infinite Domain)

10. Typically describe time evolution with no steady state.
Hyperbolic PDES
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.

11. Finite difference method for wave equation
Choose step size h in space and k in time k t x h

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

13. Finite difference method for wave equation
Choose step size h in space and k in time
Solve for ui,j+1

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

15. Finite difference method for wave equation
Can’t use this for first time step.
U at initial time given by initial condition.
ui,0 = f(xi) k t x h 15

16. Finite difference method for wave equation
Use initial derivative to make first time step.
U at initial time given by initial condition k t x h 16

17. Finite difference method for wave equation
Which discrete values influence ui,j+1 ? k t x h

18. Finite difference method for wave equation
Which discrete values influence ui,j+1 ? k t x h

19. Finite difference method for wave equation
Which discrete values influence ui,j+1 ? k t x h

20. Finite difference method for wave equation
Which discrete values influence ui,j+1 ? k t x h

21. Finite difference method for wave equation
Which discrete values influence ui,j+1 ? k t x h

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

23. 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.

24. CFL (Courant, Friedrichs, Lewy) Condition
Unstable: part of domain of dependence of PDE is outside discrete domain of dependence
x-ct=constant
x+ct=constant k t x h

25. CFL (Courant, Friedrichs, Lewy) Condition
Possibly stable: domain of dependence of PDE is inside discrete domain of dependence
x+ct=constant
x-ct=constant k t x h

26. CFL (Courant, Friedrichs, Lewy) Condition
Boundary of unstable: domain of dependence of PDE is discrete domain of dependence
x+ct=constant
x-ct=constant k t x h

27. CFL (Courant, Friedrichs, Lewy) Condition
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

28. 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.

29. 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.

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

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

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

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