Modeling Tools for Environmental Engineers and Scientists

2∂ f f i1,j1 – f i–1,j1 – f i1,j–1 f i–1,j–1 (3.35)∂ x∂y4h 2Worked Example 3.6The advective-diffusive transport of a pollutant in a river, undergoing afirst-order decay reaction, can be modeled by the following equation (asderived in Chapter 2):2 ∂ C = E ∂ C∂t∂t2 – U ∂ C – kC∂xwhere C is the concentration of the pollutant, E is the dispersion coefficient,U is the velocity, k is the reaction rate constant, t is the time, and x is the distancealong the river. Develop the finite difference formulation to solve theabove PDE numerically.SolutionThe grid notation indicated in Figure 3.12 can be adapted as shown inFigure 3.13 for this problem. Each of the terms in the PDE can now beexpressed in the finite difference form as follows: ∂ C C i,n1 – C i,n∂thE ∂ 2C∂x2 E C i+1,n – 2C i,n + C i–1,n h2U ∂ C i,n – C i–1,n U∂xC hkC k C i,n C i,n+12 Hence, combining all of the above, the finite difference form of the PDE is C i,n1 – C i,nhC i+1,n – 2C i,n + C i–1,n = E – U C i,n –h2– k C i,n C i,n+12 C i–i,nh © 2002 by CRC Press LLC