Numerical Simulation of a One-Dimensional Water-Quality Model in a Stream Using a Saulyev Technique with Quadratic Interpolated Initial-Boundary Conditions

and Applied Analysis 3 and boundary conditions are c (0, t) = g (t) , 0 ≤ t ≤ T, c (L, t) = h (t) , 0 ≤ t ≤ T, (3) where x is the longitudinal distance along the stream, t is time, T is the last time, and f(x), g(t), and h(t) are interpolated functions, while c(x, t) is the concentration averaged in depth at the point x and at time t, u(x, t) is the water flow velocity in the xdirection for all x ∈ [0, L] at time t,D is the dispersion coefficient, andK is themass decay rate. 3. Numerical Technique 3.1. An Explicit Finite Difference Technique. e solution domain of the problem is covered by a mesh of grid lines. e grid point (xi, tn) is defined by xi = iΔx for all i = 0, 1, 2, . . . ,M and tn = nΔt for all n = 0, 1, 2, . . . , N in which M andN are positive integers, where xi and tn are parallel to the space and time coordinate axes. e constant spatial and temporal grid spacing are Δx = L/M and Δt = T/N. Consider the following approximations of the derivative in the advection-diffusion equation which incorporate time weights θ as follows [10]: ∂c ∂t = cn+1 i − cn i Δt , u ∂c ∂x = un+1 i ( 1 2Δx) [cn+1 i − cn+1 i−1 + cn i+1 − cn i ] , D ∂2c ∂x2 = D( θ (Δx)2) [c n+1 i−1 − cn+1 i + cn i+1 − cn i ] + D((1 − θ) (Δx)2 ) [c n i+1 − 2cn i + cn i+1] , (4) where θ is the weighting factor. Substituting (4) into (1), we get [10] [1 + Cr 2 + θ ( Cr Pe )] cn+1 i − [Cr 2 + θ ( Cr Pe )] cn+1 i−1 = (Cr Pe − θ (Cr Pe )) cn i−1 + [1 + Cr 2 − 2 ( Cr Pe ) + θ (Cr Pe ) + KΔt] cn i + [Cr Pe − Cr 2 ] cn i+1 (5) for 1 ≤ i ≤ M − 1 and 1 ≤ n ≤ N − 1, where Cr = uΔt/Δx is Courant number and Pe = uΔx/D is Peclet number. Although (5) does not seem explicit, because cn+1 i−1 and cn+1 i are on the le -hand side, a suitable use of the equation makes it explicit. erefore, (5) can be written in the following form: cn+1 i = 1 [1 + Cr/2 + θ (Cr/Pe)] {[ Cr 2 + θ ( Cr Pe )] cn+1 i−1 + [Cr Pe − θ (Cr Pe )] cn i−1 + [1 + Cr 2 − 2 ( Cr Pe ) + θ (Cr Pe ) + KΔt] cn i + [Cr Pe − Cr 2 ] cn i+1} . (6) For the advection term, we apply the scheme of either u > 0 or u < 0. us, the scheme is restricted to single-direction velocity fields, with c being transported from le to right by the flow, so the Saulyev scheme is the appropriate choice for discretizing the advective term. In (6), the term at time level n + 1, cn+1 i−1 , has already been computed at spatial point i − 1 by marching in the direction of increasing i. is scheme is an explicit finite difference method. In this case, only a single value, cn+1 i , will be unknown. is scheme is known as Saulyev’s formula and the main advantage of it is that it is unconditionally stable and explicit [10]. 3.2. Iterative Method for the Initial and Boundary Conditions


Introduction
Water quality must be protected and maintained for several uses, the principal ones being domestic water supply, energy production, industry, agriculture, fish, and wildlife.Mathematical modeling of the water pollution measurement and control in the water area has been examined.In [ ], a simulation process showing that water pollution levels can be reduced to an agreed standard at the lowest cost is proposed.
In [ ], mathematical modeling of the transport salinity, pollutants, and suspended matter in shallow water that involves the numerical solution of an advection-diffusion equation is proposed.A novel technique of finite difference methods is proposed.In [ ], the authors also proposed a mathematical modeling of the transport salinity, pollutants, and suspended matter in shallow water that involves the numerical solution of an advection-diffusion equation in the technique of flux-corrected scheme of finite difference methods.It is available for the solution of the depth-integrated form of the advection-diffusion equation.In [ , ], the advection and diffusion terms are solved by two different numerical methods.
In [ ], the authors used a weighted discretization method with the modified equivalent partial differential equation for solving the one-dimensional advection-diffusion equation.In [ ], the authors introduce the central difference approximation that gives some negative concentration in the neighboring cell due to a large advection flux.In [ ], the authors proposed a numerical dispersion by introducing an upstream interpolation method, namely, QUICK (Quadratic Upstream Interpolation Convective Kinematics), for one-dimensional unsteady flow.In [ ], parabolic partial differential equations with a nonstandard initial condition, featured in the mathematical modeling of many phenomena, are proposed.Saulyev's explicit schemes are an economical implement to use.ese unconditionally explicit schemes are very simple to program and compute.e new explicit schemes developed are very efficient and they need less CPU time than the implicit methods.e explicit finite difference schemes are very easy to implement for similar higherdimensional problems.In [ ], a user friendly and a flexible solution algorithm are proposed for the numerical solution of the one-dimensional advection-diffusion equation (ADE), and an explicit spreadsheet simulation (ESS) technique is used instead of a computer code.In the numeric solution of ADE using finite differences, either a small value of the Courant number such as .-. is used for oscillationfree results or an artificial diffusion is used in order to reduce oscillation.In order to provide for small Courant numbers, it is necessary to choose a small time step and/or grid size; however, this increases the computation time.While the proposed ADEESS solution technique uses an unconditional stable Saulyev scheme, it gives highly accurate results even for the values of the Courant numbers as high as -.By varying only the values of the temporal weighted parameter (), namely, , ., and , respectively, the problems are solved.
e model results for the value of  = 0 appear to be in good agreement with the analytical solutions.
In [ ], a better finite difference scheme to solve the dynamic one-dimensional advection-dispersion-reaction equations (ADRE) is focused upon, and the effect of nonuniform water flows in a stream is considered.ere are two mathematical models used to simulate pollution due to sewage effluent.e first model is a hydrodynamic model for numerical techniques.
e Crank-Nicolson method is used to approximate the solution.e second model is an advection-dispersion-reaction model; the explicit schemes are introduced.e revised explicit schemes are modified from two computation techniques of uniform flow stream problems: forward time central space (FTCS) and Saulyev schemes for the dispersion model.A comparison of both schemes regarding the stability aspect is provided so as to illustrate their applicability to the real-world problem.
e dispersion model provides the pollutant concentration field.In [ ], a modified MacCormack method is subsequently employed in the dispersion model.e proposed method is a simply remarkable alteration to the MacCormack method so as to make it more accurate without any significant loss of computational efficiency.e results obtained indicate that the proposed modified MacCormack scheme does improve the prediction accuracy compared to the traditional MacCormack method.In [ ], the authors proposed a simple revision to the MacCormack and Saulyev schemes that improves their accuracy for high Peclet number problems, which are named the Saulyev and MacCormack schemes, respectively, greatly improving the prediction accuracy over the original ones.ey proposed a new scheme that guarantees the positivity of the solutions for arbitrary step sizes.In [ ], they developed a numerical technique to approximate the solution of an advection-diffusion-reaction equation in one spatial dimension with constant velocity and diffusion.In [ ], the Preissmann four-point partialnode implicit scheme is used to solve a one-dimensional hydrodynamic and water-quality model.In [ ], a nondimensional form of a two-dimensional hydrodynamic model with a generalized boundary condition and initial conditions for describing the elevation of water wave in an open uniform reservoir is proposed.e separation of variables method with mathematical induction is employed to find an analytical solution to the model.In [ ], the traditional Crank-Nicolson method is also used in the hydrodynamic model.At each step, the flow velocity fields calculated from the hydrodynamic model are the inputs into the waterquality model.A new fourth-order scheme and a Saulyev scheme are simultaneously employed in the water-quality model.In [ ], the hydrodynamics model coupled with water quality is established by MIKE FM so ware to simulate the current situation of Erhai Lake.
e water quality is also simulated by the two-dimensional hydrodynamics and waterquality coupled model.e simple explicit schemes have the advantages of simplicity in computing without losing more accuracy and these schemes are precedent for several model applications.To identify the best one of these simple schemes, comparative studies of these are necessary.
e collected field data is not suitable to input into a mathematical model.e data is varied by time.e timedependent distributions of discharged pollutant concentration and water flow velocity are required.It is complicated work if we input them into computer implementation while a given function has a simpler operation.e object of this research is to propose an interpolation technique to all of the collected field data such as water pollutant concentration at the released polluted water point and the water flow velocity along the considered water stream.e revision shows good agreement solutions.e proposed technique is suitable to be used in several real-world problems because it is easy to program and because of the straightforwardness of the implementation.According to field water-quality data, the data will be implemented to be a function of the boundary condition.
e Lagrange interpolation technique is used to synthesize their boundary conditions as required.A simple advectiondiffusion-reaction numerical simulation is proposed using the Saulyev scheme.e proposed numerical technique uses an unconditionally stable method.A large or small time step and/or grid size can be employed in the proposed techniques.We apply the method to two problems with different data for obtaining the right and le boundary conditions.e results of the model show that the calculated results are reliable approximations..In this section, we consider the parabolic equation.e mathematical model describing the transport and diffusion processes is a one-dimensional advection-diffusion-reaction equation (ADRE):

Initial and Boundary Conditions.
e initial condition is Abstract and Applied Analysis and boundary conditions are where  is the longitudinal distance along the stream,  is time,  is the last time, and (), (), and ℎ() are interpolated functions, while (, ) is the concentration averaged in depth at the point  and at time , (, ) is the water flow velocity in the direction for all  ∈ [0, ] at time ,  is the dispersion coefficient, and  is the mass decay rate.

An Explicit Finite Difference
Technique.e solution domain of the problem is covered by a mesh of grid lines.
Consider the following approximations of the derivative in the advection-diffusion equation which incorporate time weights  as follows [ ]: where  is the weighting factor.Substituting ( ) into ( ), we get [ ] erefore, ( ) can be written in the following form: For the advection term, we apply the scheme of either  > 0 or  < 0. us, the scheme is restricted to single-direction velocity fields, with  being transported from le to right by the flow, so the Saulyev scheme is the appropriate choice for discretizing the advective term.In ( ), the term at time level  + 1,  +1 −1 , has already been computed at spatial point  − 1 by marching in the direction of increasing . is scheme is an explicit finite difference method.In this case, only a single value,  +1  , will be unknown.is scheme is known as Saulyev's formula and the main advantage of it is that it is unconditionally stable and explicit [ ].
e Taylor polynomials agree as closely as possible with a given function at a specific point, but they concentrate their accuracy near that point.A good interpolation polynomial needs to provide a relatively accurate approximation over an entire interval, and Taylor polynomials do not generally do this.e Taylor polynomials are [ ] For the Taylor polynomials, all the information used in the approximation is concentrated at the single number  0 , so these polynomials will generally give inaccurate approximations as we move away from  0 .is limits Taylor polynomial approximation to the situation in which approximations are needed only at numbers close to  0 .For ordinary computational purposes, it is more efficient to use methods that include information at various points.e primary use of Taylor polynomials in numerical analysis is not for approximation purposes, but for the derivation of numerical techniques and error estimation.

Lagrange Interpolating
Polynomials.e problem of determining a polynomial of degree one that passes through the ( 0 ,  0 ) distinct points ( 0 ,  0 ) and ( 1 ,  1 ) is the same as approximating a function  for which ( 0 ) =  0 and ( 1 ) =  1 by means of a first-degree polynomial interpolation, or agreeing with the values of  at the given points.Using this polynomial for approximation within the interval given by the endpoints is called polynomial interpolation.Define the functions ( ) e linear Lagrange interpolating polynomial through ( 0 ,  0 ) and ( 1 ,  1 ) is en,  is the unique polynomial of degree at most one that passes through ( 0 ,  0 ) and ( 1 ,  1 ).In this case, we first construct, for each  = 0, 1, . . ., , a function  , () with the property that  , (  ) = 0, when  ̸ =  and  , (  ) = 1.To satisfy  , (  ) = 0 for each  ̸ = , it is required that the numerator of  , () contains the term To satisfy  , (  ) = 1, the denominator of  , () must be this same term but evaluated at  =   .us, where, for each  = 0, 1, . . ., , ( ) We will write  , () simply as   () when there is no confusion as to its degree.
where () is the interpolating polynomial given in ( 13).The error formula in ( 15) is an important theoretical result because Lagrange polynomials are used extensively in numerical differentiation and integration methods.
e error in applied mathematics is the difference between a true value and an estimate, or the approximation of that value.In numerical analysis, round-off error is exemplified by the difference between the true values of the irrational number.
e approximation error in some data is the discrepancy between an exact value and some approximation to it.An approximation error can occur because the measurement of the data is not precise because instruments and approximations are used instead of the real data.In ( ), it is implied that the error in linear interpolation is |() − ()|, where f() is the interpolating polynomial.
Interpolating the th Lagrange interpolation polynomial can be described in a simpler form as   ().It is difficult to interpolate a river channel, because it has unknown functions of initial conditions and boundary conditions.e interpolation of field data uses ( ) for interpolating ( ).Use every nodes  0 ,  1 , . . .,   that are distinct numbers in the interval [, ] by an iterative explicit finite difference technique to find the second Lagrange interpolation polynomial for () ≈ (), where () is the interpolating polynomial [ ].

Numerical Experiments
Suppose that the measurement of pollutant concentration  in a nonuniform flow stream is aligned with longitudinal distance, .(km) total length and .(m) depth.ere is a plant which discharges wastewater into the stream and the pollutant concentrations at the discharge point are (0, ) = () (mg/L) and (1, ) = ℎ() (mg/L) at 0 ≤  ≤ 1 for all  > 0 and (, 0) = () (mg/L) at  = 0. e analytical solution to Abstract and Applied Analysis

Discussion
In this research, the approximation of the pollutant concentrations of a simple advection-diffusion reaction numerical simulation using the Saulyev schemes is shown in Tablesand Figures -. e  gives a smooth solution compared to other values.Increasing the mass decay rate affects the maximum concentration level.
e interpolation results must be crude mesh as field data.e numerical results can be fine mesh or crude mesh.In  is used to predict their initial and boundary conditions as needed.

Conclusion
e proposed Saulyev finite difference scheme with the quadratic interpolation to the initial-boundary conditions technique is an unconditionally stable finite difference method.A large or small time step and/or grid size can be employed in the proposed techniques.
e numerical experiment shows that the calculated results are reasonable approximations.e revision shows good agreement solutions.e proposed interpolation technique is suitable to be used in the real-world problem because it is easy to computer-code and because of the straightforwardness of the computer implementation.According to the collected waterquality data, functions that satisfy boundary conditions must be implemented.e computed results are verified by the numerical accuracy.e proposed technique gives reliable solutions to these processes.
Theorem 3 (see [ ]).Suppose  0 ,  1 , . . .,   are distinct numbers in the interval [, ] and  ∈  +1 [, ].Then, for each  in [, ], a number () (generally unknown) between  0 ,  1 , . . .,   , and hence in (, ), exists with Table and Figures -, we can see that the maximum errors of approximated pollutant concentration are reducing while the Peclet numbers are decreased.e maximum error of analytical and interpolation technique is shown in Table ; the right boundary condition is 3.1640 × 10 −08 , the le boundary condition is 3.3258 × 10 −08 , and the initial condition is 2.836181 × 10 −10 .Comparison of the analysis and interpolation technique is shown in Figures -. : e maximum error of approximated pollutant concentration at  = 0.25, 0.50, and 0.75, for all  ∈ [0, 1]. T