Unconditional stable numerical techniques for a water-quality model in a non-uniform flow stream
- Nopparat Pochai^{1, 2}Email author
https://doi.org/10.1186/s13662-017-1338-4
© The Author(s) 2017
Received: 1 February 2017
Accepted: 26 August 2017
Published: 13 September 2017
Abstract
Two mathematical models are used to simulate water quality in a non-uniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the water elevation. The second model is an advection-diffusion-reaction model that provides the pollutant concentration field. Both models are formulated as one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the inputs into the second model. A new fourth-order scheme and a Saulyev scheme are simultaneously employed in the second model. This paper proposes a remarkably simple alteration to the fourth-order method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed new fourth-order scheme, coupled to the Saulyev method, does improve the prediction accuracy compared to that of the traditional methods.
Keywords
MSC
1 Introduction
The pollution levels in a stream can be measured via data collection. This is rather difficult and complex, and the results obtained deviate in measurement from one point in each time/place to another when the water flow in the stream is not uniform. In water quality modeling for a non-uniform flow stream, the governing equations used are the hydrodynamic model and the dispersion model. The one-dimensional shallow water equation and the advection-dispersion-reaction equation govern the first and the second models, respectively.
Numerous numerical techniques for solving such models are available. In [1], a finite element method for solving a steady water pollution control to achieve a minimum cost is used. Numerical techniques for solving the uniform flow of a stream water quality model, especially the one-dimensional advection-dispersion-reaction equation, are presented in [2–5], and [6].
The non-uniform flow model requires the velocity of the current at any point and any time in the domain. A hydrodynamic model provides the velocity field and tidal elevation of the water. In [7–9], and [10], the hydrodynamic model and the advection-dispersion equation are used to approximate the velocity of the water current in a bay, uniform reservoir and stream, respectively. Among these numerical techniques, finite difference methods, including both explicit and implicit schemes, are mostly used for one-dimensional domains, such as longitudinal stream systems [11, 12].
There are two mathematical models used to simulate water quality in non-uniform water flow systems. The first is the hydrodynamic model. This provides the velocity field and the water elevation. The second is the dispersion model. This gives the pollutant concentration field. The traditional Crank-Nicolson method is used in the hydrodynamic model. At each step, the calculated flow velocity fields of the first model are inputs into the second model [9, 10, 13].
Numerical techniques to solve the non-uniform flow of stream water containing one-dimensional advection-dispersion-reaction equation are presented in [10]. The fully implicit scheme (the Crank-Nicolson method) is used to solve the hydrodynamic model and the backward time-central space (BTCS) for the dispersion model. In [13], the Crank-Nicolson method is also used to solve the hydrodynamic model while the explicit Saulyev scheme is used to solve the dispersion model.
Research on finite difference techniques for the dispersion model have concentrated on computation accuracy and numerical stability. Many complicated numerical techniques, such as the QUICK scheme, the Lax-Wendroff scheme, and the Crandall scheme, have been studied to increase performance. These techniques focus on their advantages in terms of stability and higher-order accuracy [3].
Simple finite difference schemes are becoming more attractive for model use. Simple explicit methods include the forward time-central space (FTCS) scheme, the MacCormack scheme, and the Saulyev scheme. Implicit schemes include the BTCS and the Crank-Nicolson scheme [12]. These schemes are either first-order or second-order accurate and have advantages in programming and computing without losing much accuracy, thus they are used for many model applications [3].
A third-order upwind scheme for the advection-diffusion equation using a simple spreadsheet simulation is proposed in [14]. In [15], a new flux splitting scheme is proposed. The scheme is robust and converges as fast as the Roe splitting scheme. The Godunov mixed method for advection-dispersion equations is introduced in [16]. A time-splitting approach for the advection-dispersion equations is also considered. In addition, [17] proposes time-split methods for multi-dimensional advection-diffusion equations in which advection is approximated by a Godunov-type procedure, and diffusion is approximated by a low-order mixed finite element method. In [18], a flux-limiting solution technique for the simulation of a reaction-diffusion-convection system is proposed. A composite scheme to solve scalar transport equations in a two-dimensional space, that accurately resolves sharp profiles in the flow, is introduced. The total variation diminishing implicit Runge-Kutta method for dissipative advection-diffusion problems in astrophysics is proposed in [19]. They derive dissipative space discretizations and demonstrate that, together with specially adapted total-variation-diminishing or strongly stable Runge-Kutta time discretizations with adaptive step-size control, these yield reliable and efficient integrators for the underlying multi-dimensional non-linear evolution equations.
We propose simple revisions to a new fourth-order scheme that improve its accuracy for the problem of water quality measurement in a non-uniform water flow in a stream. In the following sections, the formulation of a new fourth-order scheme is introduced. The proposed revision of a new fourth-order scheme with the Saulyev method is described.
The result from the hydrodynamic model is the water flow velocity used in the advection-dispersion-reaction equation to determine the pollutant concentration field. Friction forces, due to the drag of the sides of the stream, are considered. The theoretical solution to the model at the end point of the domain that guarantees the accuracy of the approximate solution is presented in [9, 10], and [13].
2 Model formulation
2.1 The hydrodynamic model
2.2 Dispersion model
3 Crank-Nicolson method for the hydrodynamic model
4 A new fourth-order scheme with a Saulyev method for the advection-dispersion equation
4.1 The employment of a Saulyev method to the left and the right boundary conditions
5 Numerical experiment
The error defined by \(\pmb{\operatorname{error}(T_{\mathrm{mxe}}) = \max \vert C(0.50,t)-\widetilde{C}(0.5,t) \vert }\) for all \(\pmb{0\leq t \leq 5}\) , for some \(\pmb{T_{\mathrm{mxe}}\in [0,5]}\) , where C and \(\pmb{\widetilde{C}}\) are theoretical solutions and new fourth-order scheme solutions, respectively
Δ x | Δ t | C (0.5, t ) | \(\boldsymbol{\widetilde{C}(0.5,t)}\) | \(\boldsymbol{\operatorname{error}(T_{\mathrm{mxe}})} \) | \(\boldsymbol{T_{\mathrm{mxe}}}\) |
---|---|---|---|---|---|
0.0500 | 0.002500 | 0.9105 | 1.0961 | 0.1856 | 2.2775 |
0.0500 | 0.001250 | 0.9095 | 1.0889 | 0.1794 | 2.2738 |
0.0500 | 0.000625 | 0.9090 | 1.0654 | 0.1764 | 2.2719 |
0.0125 | 0.000625 | 0.5366 | 0.1289 | 0.4076 | 1.9513 |
0.0250 | 0.000625 | 0.0678 | 0.2856 | 0.2178 | 1.6856 |
0.1000 | 0.000625 | 0.9908 | 1.1121 | 0.1213 | 2.5000 |
The comparison sensitivity to discretization sizes of approximated pollutant concentrations while time discretizations are halved
Δ x | Δ t | \(\boldsymbol{\Delta t/(\Delta x)^{2}}\) | C (1.0,40) |
---|---|---|---|
0.0500 | 0.002500 | 1.00 | 0.70470 |
0.0500 | 0.001250 | 0.50 | 0.75667 |
0.0500 | 0.000625 | 0.25 | 0.76400 |
The proposed technique gives the accurate results that depend on some spatial discretization sizes. It is a remarkably simple alteration to the fourth-order method so as to make it more accurate without any significant loss of computational efficiency.
6 Application to a non-uniform flow stream water quality assessment
Consider the measurement of a pollutant concentration C in a non-uniform flow stream. The stream is aligned with a longitudinal distance of 1.0 (km) and a depth of 1.0 (m). There is a plant which discharges waste water into the stream and the pollutant concentration at the discharge point is \(C(0,t) = C_{0} = 1\) (mg/L) at \(x=0\) for all \(t>0\), there is no rate of change of pollutant level \(\frac{\partial C}{\partial x}=0\) at \(x=1.0\) for all \(t>0\), and there is no initial pollutant \(C(x,0) = 0 \) (mg/L) at \(t=0\). The elevation of water at the discharge point can be described as a function \(d(0,t) = f(t) = \sin t\) (m) for all \(t>0\), and the elevation does not change at \(x = 1.0\) (km). The physical parameter of the pollutant matter is a diffusion coefficient \(D=0.1\) (m^{2}/s).
The velocity of water flow \(\pmb{u(x,t)}\) , \(\pmb{\Delta x = 0.05}\) , \(\pmb{\Delta t = 0.00125}\)
t , x | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 0.5421 | 0.5505 | 0.5421 | 0.5202 | 0.4859 | 0.4362 | 0.3701 | 0.2901 | 0.1998 | 0.1029 | 0.0000 |
20 | 1.3284 | 1.2315 | 1.1228 | 1.0041 | 0.8766 | 0.7416 | 0.6001 | 0.4538 | 0.3042 | 0.1526 | 0.0000 |
30 | 0.3291 | 0.2628 | 0.2058 | 0.1578 | 0.1182 | 0.0861 | 0.0606 | 0.0405 | 0.0246 | 0.0116 | 0.0000 |
40 | −1.1343 | −1.0752 | −0.9995 | −0.9086 | −0.8044 | −0.6884 | −0.5627 | −0.4290 | −0.2894 | −0.1457 | 0.0000 |
The pollutant concentration \(\pmb{C(x,t)}\) using the new fourth-order scheme and employing Saulyev scheme, Δ x = 0.05, Δ t = 0.00125
t , x | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|---|---|---|---|---|
10 | 1.00000 | 0.98707 | 0.96381 | 0.92269 | 0.85430 | 0.75311 | 0.62458 | 0.48989 | 0.38012 | 0.31931 | 0.30376 |
20 | 1.00000 | 0.99226 | 0.97058 | 0.92379 | 0.84269 | 0.72904 | 0.60095 | 0.48632 | 0.40726 | 0.36902 | 0.35910 |
30 | 1.00000 | 0.81678 | 0.67155 | 0.60392 | 0.58238 | 0.57723 | 0.57518 | 0.57294 | 0.57033 | 0.56750 | 0.56423 |
40 | 1.00000 | 0.73266 | 0.71148 | 0.71812 | 0.71721 | 0.71552 | 0.71391 | 0.71215 | 0.71013 | 0.70778 | 0.70470 |
7 Conclusions
In this paper, the unconditionally stable Crank-Nicolson method is proposed to solve a one-dimensional hydrodynamic model with damped force due to the drag of stream sides. The one-dimensional advection-diffusion equation in a non-uniform flow in the stream is solved by using a new fourth-order scheme employing the unconditionally stable Saulyev method near the left and right boundary conditions. The new fourth-order scheme gives accurate results without any significant loss of computational efficiency. The results obtained indicate that the proposed new fourth-order scheme, coupled with the unconditionally stable Saulyev method, improves the prediction accuracy compared to that of the traditional computation techniques.
Declarations
Acknowledgements
This work is supported by the Thailand Research Fund (TRG5780016) and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The author greatly appreciates valuable comments received from the referees.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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