### The hydrodynamic model

In this section, we derive a simple hydrodynamic model describing the water current and elevation by the one-dimensional shallow water equation. We make the usual assumption in the continuity and momentum balance, *i.e.*, we assume that the Coriolis and shearing stresses are small, and the surface wind is soft [7, 9, 10, 20]. We obtain the one-dimensional shallow water equations

$$\begin{aligned} &\frac{\partial \zeta}{\partial t} + \frac{\partial }{\partial x} \bigl[(h+ \zeta)u\bigr] = 0, \end{aligned}$$

(1)

$$\begin{aligned} &\frac{\partial u}{\partial t} + g \frac{\partial \zeta}{\partial x} = 0, \end{aligned}$$

(2)

where *x* is the longitudinal distance along the stream (m), *t* is time (s), \(h(x)\) is the depth measured from the mean water level to the stream bed (m), \(\zeta(x,t)\) is the elevation from the mean water level to the temporary water surface or the tidal elevation (m/s), and \(u(x,t)\) is the velocity components (m/s), for all \(x \in [0,l]\).

Assuming that *h* is a constant and that \(\zeta \ll h\), equations (1) and (2) reduce to

$$\begin{aligned} &\frac{\partial \zeta}{\partial t} + h\frac{\partial u}{\partial x} \doteq 0, \end{aligned}$$

(3)

$$\begin{aligned} &\frac{\partial u}{\partial t} + g \frac{\partial \zeta}{\partial x} = 0. \end{aligned}$$

(4)

We obtain a dimensionless equation by letting \(U = u / \sqrt{gh}\), \(X = x/l\), \(Z = \zeta / h\) and \(T = t \sqrt{gh}/l\). Substituting these expressions into equations (3) and (4) leads to

$$\begin{aligned} &\frac{\partial Z}{\partial T} + \frac{\partial U}{\partial X} = 0 , \end{aligned}$$

(5)

$$\begin{aligned} &\frac{\partial U}{\partial T} + \frac{\partial Z}{\partial X} = 0. \end{aligned}$$

(6)

In [9, 10], and [13], a damping term is introduced into equations (5) and (6) to represent the frictional forces due to the drag of sides of the stream. We have

$$\begin{aligned} &\frac{\partial Z}{\partial T} + \frac{\partial U}{\partial X} = 0, \end{aligned}$$

(7)

$$\begin{aligned} &\frac{\partial U}{\partial T} + \frac{\partial Z}{\partial X} = -U. \end{aligned}$$

(8)

The initial conditions at \(t=0\) and \(0\leq X\leq 1\) are \(Z= 0\) and \(U=0\). The boundary conditions for \(t>0\) are \(Z = e^{it}\) at \(X=0\) and \(\frac{\partial Z}{\partial X} =0\) at \(X=1\). Equations (7) and (8) are called the damped equations. We solve the damped equations using a finite difference method in \([0,1]\times [0,T]\). Since *Z* may be used to represent the vertical coordinate, *U* may be used to represent the approximated solutions, and *T* may be used to represent the time at which the maximum error of computed solutions is found, it is convenient to use \(u, d, t\) and *x* for \(U, Z, T\) and *X*, respectively. We have

$$\begin{aligned} &\frac{\partial u}{\partial t} + \frac{\partial d}{\partial x} = -u, \end{aligned}$$

(9)

$$\begin{aligned} &\frac{\partial d}{\partial t} + \frac{\partial u}{\partial x} = 0. \end{aligned}$$

(10)

The initial conditions are \(u=0\) and \(d=0\) at \(t=0\). The boundary conditions are \(d(0,t)=f(t)\) and \(\frac{\partial d}{\partial x}=0\) at \(x=1\).

### Dispersion model

In a stream water quality model, the governing equation is the dynamic one-dimensional advection-dispersion equation. A simplified representation, averaging the equation over the depths, as shown in [2–4, 10], and [6], is

$$ \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} = D\frac{\partial^{2} C}{\partial x^{2}}. $$

(11)

Here \(C(x,t)\) is the concentration averaged over the depth at the point *x* at time *t* (mg/L), *D* is the diffusion coefficient (m^{2}/s), and \(u(x,t)\) is the velocity component (m/s), for all \(x \in [0,L]\). The initial conditions and the left boundary conditions are usually determined by observations. The initial pollutant concentration is \(C(x,0) = C_{0}\) at \(t=0\) for all \(x>0\), where \(C_{0}\) is a positive constant. The released pollutant concentration on the left boundary condition is given by \(C(0,t) = r(t)\) at \(x=0\), where \(r(t)\geq0\). The observed rate of change of the pollutant concentration on the right boundary is assumed to be a constant \(\frac{\partial C}{\partial x} = S_{0}\) at \(x=L\), where \(S_{0}\) is an arbitrary constant.