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Numerical Integration
CE5504 – Surface Water Quality Modeling
non-idealized loading functions
variable parameters
multi-segment systems
non-linear kinetics
The Euler Method
CE5504 – Surface Water Quality Modeling
dXX
dt
dX X dt
new oldX X dX
The Effect of Step Size
CE5504 – Surface Water Quality Modeling (Spain 1982, Figure 5.1)
Code
CE5504 – Surface Water Quality Modeling
max0For t to t
dX X dt
X X dX
Next t
max0For t to t Step dt
dX X dt
X X dX
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max0
,
For t to t Step dt
dX X dt
X X dX
Print t X
Next t
Code (continued)
CE5504 – Surface Water Quality Modeling
max0
1/
,
For t to t
For k dt
dX X dt
X X dX
Next k
Print t X
Next t
Advanced Numerical TechniquesThe Heun’s Method
CE5504 – Surface Water Quality Modeling
For i = 0 To tmax
For j = 1 To 1 / dt
k1 = mu * x X1 = x + dt * k1
k2 = mu * X1
x = x + (k1 + k2) / 2 * dt Next j Next i
CE5504 – Surface Water Quality Modeling
Advanced Numerical TechniquesThe 4th Order Runge Kutta Method
For i = 0 To tmax
For j = 1 To 1 / dt
k1 = mu * x X1 = x + 0.5 * dt * k1
k2 = mu * X1
X2 = x + 0.5 * dt * k2 k3 = mu * X2 X3 = x + dt * k3 k4 = mu * X3 x = x + (k1 + 2 * k2 + 2 * k3 + k4) / 6 * dt Next j Next i
Advanced Numerical TechniquesError Comparison
CE5504 – Surface Water Quality Modeling
Error (%) for various values of dt at t = 5 days
dt Euler Heun 4th RK
1 98.90 85.81 23.70
0.5 95.35 56.70 3.60
0.1 58.68 5.58 0.01
0.01 9.40 0.07
0.001 0.99 0.001
0.0001 0.10
0.00001 0.02
0.000001 0.001
Error criterion: <=0.01 %
