^{1, 2}

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^{2}

The stream water quality model of water quality assessment problems often involves numerical methods to solve the equations. The governing equation of the uniform flow model is one-dimensional advection-dispersion-reaction equations (ADREs). In this paper, a better finite difference scheme for solving ADRE is focused, and the effect of nonuniform water flows in a stream is considered. Two mathematical models are used to simulate pollution due to sewage effluent. The first is a hydrodynamic model that provides the velocity field and elevation of the water flow. The second is a advection-dispersion-reaction model that gives the pollutant concentration fields after input of the velocity data from the hydrodynamic model. For numerical techniques, we used the Crank-Nicolson method for system of a hydrodynamic model and the explicit schemes to the dispersion model. The revised explicit schemes are modified from two computation techniques of uniform flow stream problems: forward time central space (FTCS) and Saulyev schemes for dispersion model. A comparison of both schemes regarding stability aspect is provided so as to illustrate their applicability to the real-world problem.

Field measurement and mathematical simulation are methods to detect the amount of pollutant in water area. For the shallow water mass transport problems that presented in [

The most of nonuniform flow model requires data concerned with velocity of the current at any point and any time in the domain. The hydrodynamics model provides the velocity field and tidal elevation of the water. In [

The numerical techniques to solve the nonuniform flow of stream water quality model, one-dimensional advection-diffusion-reaction equation, is presented in [

The finite difference methods, including both explicit and implicit schemes, are mostly used for one-dimensional problems such as in longitudinal river systems [

The simple finite difference schemes become more inviting for general model use. The simple explicit schemes include Forward-Time/Centered-Space (FTCS) scheme and the Saulyev scheme. These schemes are either first-order or second-order accurate [

In this paper, we will use more economical computation techniques than the method in [

The results from hydrodynamic model are data of the water flow velocity for advection-diffusion-reaction equation which provides the pollutant concentration field. The term of friction forces due to the drag of sides of the stream is considered. The theoretical solution of the model at the end point of the domain that guaranteed the accurate of the approximate solution is presented in [

The stream has a simple one-space dimension as shown in Figure

The shallow water system.

The continuity and momentum equations are governed by the hydrodynamic behavior of the stream. If we average the equations over the depth, discarding the term due to Coriolis parameter, shearing stresses, and surface wind [

Assume that

In a stream water quality model, the governing equations are the dynamic one-dimensional advection-dispersion-reaction equations (ADREs). A simplified representation by averaging the equation over the depth is shown in [

The hydrodynamic model provides the velocity field and elevation of the water. Then the calculated results of the model will be input into the dispersion model which provides the pollutant concentration field.

We will follow the numerical techniques of [

We now discretize (

We can then approximate

Taking the forward time central space technique [

The finite difference formula (

The Saulyev scheme is unconditionally stable [

From (

It is not hard to find the analytical solution

The error defined by

20 | 0.200 | 0.05000 | 1.50158 | 1.45659 | 0.04499 |

0.100 | 0.02500 | 0.09392 | 0.38856 | 0.29465 | |

0.050 | 0.01250 | −0.25831 | −0.40244 | 0.14413 | |

0.025 | 0.00625 | −0.33323 | −0.40433 | 0.07110 | |

30 | 0.200 | 0.05000 | 1.49666 | 1.45165 | 0.04501 |

0.100 | 0.02500 | −0.09358 | −0.38821 | 0.29463 | |

0.050 | 0.01250 | 0.25798 | 0.40198 | 0.14401 | |

0.025 | 0.00625 | 0.33290 | 0.40389 | 0.07099 | |

40 | 0.200 | 0.05000 | 1.50146 | 1.45644 | 0.04502 |

0.100 | 0.02500 | 0.09324 | 0.38787 | 0.29463 | |

0.050 | 0.01250 | −0.25765 | −0.40165 | 0.14401 | |

0.025 | 0.00625 | −0.33257 | −0.40356 | 0.07099 |

Comparison of analytical solution for height of water elevation with results obtained by numerical technique at the end point of the domain.

Figure

Unfortunately, the analytical solutions of hydrodynamic model could not found over entire domain. This implies that the analytical solutions of dispersion model could not carry out at any points on the domain as well.

Suppose that the measurement of pollutant concentration ^{3}) in a uniform stream at time ^{3} at ^{3} at ^{2}/s, and a first-order reaction rate ^{−1}. In the analysis conducted in this study, meshes the stream into 20, 40, 80, and 160 elements with

The velocity of water flow

10 | −0.5478 | −0.5695 | −0.5666 | −0.5199 | −0.4711 | −0.4061 | −0.3352 | −0.2571 | −0.1747 | −0.0878 | 0.0000 |

20 | 1.3101 | 1.2213 | 1.1232 | 1.0160 | 0.8931 | 0.7504 | 0.6097 | 0.4595 | 0.3099 | 0.1554 | 0.0000 |

30 | −0.4468 | −0.3731 | −0.3085 | −0.2527 | −0.2057 | −0.1651 | −0.1252 | −0.0875 | −0.0573 | −0.0276 | 0.0000 |

40 | −1.0361 | −0.9898 | −0.9258 | −0.8459 | −0.7513 | −0.6439 | −0.5258 | −0.4004 | −0.2711 | −0.1367 | 0.0000 |

50 | 1.0939 | 0.9918 | 0.8867 | 0.7791 | 0.6700 | 0.5594 | 0.4479 | 0.3356 | 0.2233 | 0.1114 | 0.0000 |

The pollutant concentration

10 | 1.0000 | 0.3231 | 0.1154 | 0.0517 | 0.0292 | 0.0194 | 0.0141 | 0.0109 | 0.0090 | 0.0079 | 0.0075 |

20 | 1.0000 | 0.9996 | 0.9971 | 0.9873 | 0.9575 | 0.8873 | 0.7588 | 0.5774 | 0.3828 | 0.2324 | 0.1720 |

30 | 1.0000 | 0.9981 | 0.9956 | 0.9910 | 0.9830 | 0.9706 | 0.9533 | 0.9327 | 0.9122 | 0.8966 | 0.8903 |

40 | 1.0000 | 0.9260 | 0.9166 | 0.9150 | 0.9145 | 0.9141 | 0.9138 | 0.9136 | 0.9135 | 0.9134 | 0.9133 |

50 | 1.0000 | 0.9993 | 0.9966 | 0.9902 | 0.9788 | 0.9633 | 0.9468 | 0.9327 | 0.9231 | 0.9179 | 0.9162 |

The pollutant concentration

10 | 1.0000 | 0.3297 | 0.1225 | 0.0570 | 0.0327 | 0.0212 | 0.0144 | 0.0096 | 0.0059 | 0.0028 | 0.0000 |

20 | 1.0000 | 0.9995 | 0.9970 | 0.9878 | 0.9619 | 0.9020 | 0.7908 | 0.6251 | 0.4238 | 0.2136 | 0.0000 |

30 | 1.0000 | 0.9972 | 0.9919 | 0.9798 | 0.9536 | 0.9026 | 0.8136 | 0.6760 | 0.4859 | 0.2530 | 0.0000 |

40 | 1.0000 | 0.3128 | 0.2031 | 0.1624 | 0.1321 | 0.1052 | 0.0807 | 0.0581 | 0.0372 | 0.0179 | 0.0000 |

50 | 1.0000 | 0.9913 | 0.9631 | 0.8991 | 0.7879 | 0.6342 | 0.4610 | 0.2988 | 0.1694 | 0.0753 | 0.0000 |

The stability of FTCS and Saulyev schemes,

FTCS scheme | Saulyev scheme | ||||

0.0125 | 0.010000 | 3.2 | 1.0665 | Unstable | Stable |

0.005000 | 1.6 | 0.5332 | Unstable | Stable | |

0.002500 | 0.8 | 0.2667 | Unstable | Stable | |

0.001250 | 0.4 | 0.1333 | Stable | Stable | |

0.000625 | 0.2 | 0.0080 | Stable | Stable | |

0.0125 | 0.010000 | 3.2 | 1.0665 | Unstable | Stable |

0.005000 | 1.6 | 0.5332 | Unstable | Stable | |

0.002500 | 0.8 | 0.2667 | Unstable | Stable | |

0.001250 | 0.4 | 0.1333 | Stable | Stable | |

0.000625 | 0.2 | 0.0080 | Stable | Stable | |

0.0125 | 0.010000 | 3.2 | 1.0665 | Unstable | Stable |

0.005000 | 1.6 | 0.5332 | Unstable | Stable | |

0.002500 | 0.8 | 0.2667 | Unstable | Stable | |

0.001250 | 0.4 | 0.1333 | Stable | Stable | |

0.000625 | 0.2 | 0.0080 | Stable | Stable |

The water velocity

The pollutant concentration

The comparison of concentration at two different time instants of the FTCS and Saulyev methods.

The pollutant concentration

The approximation of the pollutant concentrations from the FTCS technique is shown in Tables

By Figure

In this paper, it can be combined the hydrodynamic model and the convection-diffusion-reaction equation to approximate the pollutant concentration in a stream when the current which reflects water in the stream is not uniform. The technique developed in this paper the response of the stream to the two different external inputs: the elevation of water and the pollutant concentration at the discharged point. The Saulyev technique can be used in the dispersion model since the scheme is very simple to implement. By the Saulyev finite difference formulation, we obtain that the proposed technique is applicable and economical to be used in the real-world problem as aresult of the simplicity of programming and the straight forwardness of the implementation. It is also possible to find tentative better locations and the periods of time of the different discharged points to a stream.

This paper is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The author greatly appreciates valuable comments received from the referees.