1. Discontinuous Shallow Flow
A Numerical Method for
Discontinuous Shallow Flow
Fritz R. Fiedler University of Idaho
Department of Civil Engineering

2. Model Objectives Solve 2-D hydrodynamic flow equations
surface flow, rainfall, infiltration
stiff hyperbolic equations
non-linear source term
Applications
rainfall-runoff process
wetlands
flood plains

3. Overland Flow

4. Microtopography

5. Equations

6. Numerical Challenges Non-linear hyperbolic system Strong source terms
(equations stiff when h~0)
Small depths / dry areas
Large gradients
Discontinuous flow regime

7. Vector Form

8. Vector Form

9. Approach Select basic numerical scheme
Modify basic scheme to address problem-specific challenges
Develop algorithm
Develop code
Test
Iterate? (start simple)

10. MacCormack Scheme Predictor, Backward Difference
Corrector, Forward Difference

11. Split MacCormack Scheme
Lx1 Operator:

12. Friction Slope: stiff!

13. Friction Slope:
Point-Implicit Treatment

14. Convective Acceleration Upwinding
For the Lx1 operator:
If flow is in the -x direction (j+1 to j)

15. Smoothing Function

16. Lateral Inflow and Infiltration

17. Algorithm Input Define grid Initialize Solve time loop
compute lateral inflow (Newton’s Method)
compute h, p, q
output?

18. Computer Code
do 102 j=1,Nx-1
delh1(j,k) = - dtohx*(pc(j,k)-pc(j-1,k))+dt*0.5*re(j,k)
delp1(j,k) = - D(j,k)*dtohx*
& (convacc(j,k) + g*hc(j,k)**2 - g*hc(j-1,k)**2)
& D(j,k)*dt *
& ( g*(hc(j,k)+hc(j-1,k))*(z(j,k)-z(j-1,k))/hx
& Ko*0.01*pc(j,k)/8./hc(j,k)**2
& pc(j,k)/hc(j,k)*re(j,k) )
& D(j,k)*dt/hx**2*eps1*(pc(j-1,k)-2.*pc(j,k)+pc(j+1,k))
delq1(j,k) = - dtohx*
& ( pc(j,k)*qc(j,k)/hc(j,k)
& pc(j-1,k)*qc(j-1,k)/hc(j-1,k) )
& dt/hx**2*eps1*(qc(j-1,k)-2.*qc(j,k)+qc(j+1,k))
continue

19. Comparative Numerical Examples
Dam Break Problem
Published results
Physical Model Results

20. Dam Break Problem

21. Kinematic Wave Solution

22. Microtopographic Surface

23. Overland Flow Depths

24. Flow Depths and Velocity

25. Flow Channels

26. Overland Flow Depths

27. Cumulative Infiltration

28. Simulated vs. Experimental