A Numerical Method for

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A Numerical Method for Discontinuous Shallow Flow
Fritz R. FiedlerUniversity of Idaho Department
of Civil Engineering
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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

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Overland Flow
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Microtopography
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Equations
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Numerical Challenges

• Non-linear hyperbolic system
• Strong source terms
• (equations stiff when h0)
• Small depths / dry areas
• Large gradients
• Discontinuous flow regime

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Vector Form
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Vector Form
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Approach

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

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MacCormack Scheme
Predictor, Backward Difference
Corrector, Forward Difference
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Split MacCormack Scheme
Lx1 Operator
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Friction Slope stiff!
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Friction Slope Point-Implicit Treatment
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Convective Acceleration Upwinding
For the Lx1 operator
If flow is in the -x direction (j1 to j)
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Smoothing Function
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Lateral Inflow and Infiltration
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Algorithm

• Input
• Define grid
• Initialize
• Solve
• time loop
• compute lateral inflow (Newtons Method)
• compute h, p, q
• output?

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Computer Code
do 102 j1,Nx-1 delh1(j,k) -
dtohx(pc(j,k)-pc(j-1,k))dt0.5re(j,k)
delp1(j,k) - D(j,k)dtohx
(convacc(j,k) ghc(j,k)2 - ghc(j-1,k)2)
- D(j,k)dt (
g(hc(j,k)hc(j-1,k))(z(j,k)-z(j-1,k))/hx
Ko0.01pc(j,k)/8./hc(j,k)2
pc(j,k)/hc(j,k)re(j,k) )
D(j,k)dt/hx2eps1(pc(j-1,k)-2.pc(j,k)
pc(j1,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/hx2eps1(qc(j-1,k)-2.qc(j,k)
qc(j1,k)) 102 continue
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Comparative Numerical Examples

• Dam Break Problem
• Published results
• Physical Model Results

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Dam Break Problem
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Kinematic Wave Solution
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Microtopographic Surface
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Overland Flow Depths
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Flow Depths and Velocity
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Flow Channels
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Overland Flow Depths
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Cumulative Infiltration
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Simulated vs. Experimental
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