1. Hyperbolic Equations IVP: u(x,0) = f(x), - < x <  IBVP:
u(x,0) = f(x), 0 < x <  and u(0,t) = g(t), t > 0

2. IVP

4. Method of Characteristics
Let C be a characteristic curve described by x = x(t).
On C u(x,t) = u(x(t), t).
Differentiating u along C results in the equation:

5. Method of Characteristics
a is defined as the wave speed on the curve C
The change of u with respect to t on C is zero
the first equation is solved to obtain

6. Characteristics given the initial condition
the solution to the (kinematic) wave equation is
example...

7. Courant-Friedrichs-Lewy (CFL) Condition
The numerical domain of dependence must contain the analytical domain of dependence, where the analytical domain of dependence is given by the characteristic curves.
How does this govern the relationship between the time step and space step, given the wave speed a?

8. CFL Condition

9. Hyperbolic IBVPs
Explicit methods shown previously work, with addition of the BC at x = 0
Implicit methods are unconditionally stable
Some implicit methods do not require solving a system of equations!