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

dX Xdt

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 tdX X dtX X dX

Next t

max0For t to t Step dtdX X dtX X dX

Next t

Code (continued)

CE5504 – Surface Water Quality Modeling

max0

,

For t to t Step dtdX X dtX X dXPrint t X

Next t

Code (continued)

CE5504 – Surface Water Quality Modeling

max01/

,

For t to tFor k dtdX X dtX X dX

Next kPrint 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 %