Develop algorithms for the explicit solution of the Kinematic Equations when applied to a problem of runoff from a plane surface subject to a steady rate of rainfall excess. Use forward differences in the time (t) direction and backward differences in the x direction.
The initial and boundary conditions are:
- Initial condition (t=0): q=0 and y=0 for all x
- Upstream boundary (x=0): q=0 for all t
- Downstream boundary (x=L): S0=Sf
Develop the equations for the solution q, v, y as follows:
(a) In the main body of the x-t solution space using central differences.
(b) At the upstream boundary using forward differences for the spatially variant terms.