Understanding PDE Classification and Numerical Methods

1. Introduction to PDE classification Numerical Methods for PDEs Spring 2007 Jim E. Jones • References: • Partial Differential Equations of Applied Mathematics, Zauderer • Wikopedia, Partial Differential Equation

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: Parabolic Equation (Finite Domain) Heat equation Typical Boundary Conditions x=L/2 x=0 x=-L/2

4. Example: Parabolic Equation Heat equation Typical Boundary Conditions Initial temperature profile in rod Temperatures for end of rod x=L/2 x=0 x=-L/2

5. Example: Parabolic Equation (Infinite Domain) Heat equation Dirac Delta Boundary Conditions x=0

6. Dirac Delta Function The Dirac delta function is the limit of Physically it corresponds to a localized intense source of heat

7. Example: Parabolic Equation (Infinite Domain) Heat equation Dirac Delta Boundary Conditions Solution (verify)

8. Example: Parabolic Equation (Infinite Domain) t=.1 t=.01 t=1 t=10

9. Parabolic PDES • Typically describe time evolution towards a steady state. • Model Problem: Describe the temperature evolution of a rod whose ends are held at a constant temperatures. • Initial conditions have immediate, global effect • Point source at x=0 makes temperature nonzero throughout domain for all t > 0.

10. Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions

11. Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions Solution (verify)

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

13. 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

14. Example: Hyperbolic Equation (Infinite Domain) Heat equation Boundary Conditions

15. Example: Hyperbolic Equation (Infinite Domain) t=.01 t=.1 t=1 t=10

16. 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.

17. Example: Elliptic Equation (Finite Domain) Laplace’s equation Typical Boundary Conditions W

18. The Problem PDE solution (verify)

19. Elliptic Solution

20. Elliptic PDES • Typically describe steady state behavior. • Model problem: Describe the final temperature profile in a plate whose boundaries are held at constant temperatures. • Boundary conditions have global effect

21. 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