Solving One-Dimensional Advection-Dispersion with Reaction Using Some Finite-Difference Methods

Transcription

1 Applied Mathematical Sciences, Vol., 008, no. 53, 6-68 Solving One-Dimensional Advection-Dispersion with Reaction Using Some Finite-Difference Methods H. Saberi Naafi Department of Mathematics, Faculty of Sciences Guilan University, Iran H. Hainezhad Department of Mathematics, Faculty of Sciences Payame Noor University, Iran Haniye Abstract In this paper we have solved the one-dimensional advection-dispersion equation with reaction using some finite-difference methods, namely, FTCS used by B.Ataie-Ashtiani et al. (996), (we call it FTCS-Ataie) [], Upstream, DuFort-Frankel, (3-3), and Crank-Nicolson methods. The implementation and behavior of the methods have been compared by plotting the figures. It has been shown that the Crank-Nicolson method works more accurately than the other methods. The equation is the same as the one used in the article [Journal of Contaminant Hydrology 3 (996) 49-56]. Keywords: Advection, Dispersion, Finite-difference. Introduction Advection-dispersion phenomena occur in many physical situation including the transfer of heat in fluids, flow through porous media, the spread of contaminants in fluids and in chemical separation processes, and so on. The one-dimensional advection-dispersion has been used and solved by researchers, for example, it has been used to describe thermal pollution in a river system (chaudhry et al. 983) leaching of salts in soil (Bresler, 973), flow in porous media (kumar, 983), the spread of pollutants in streams (Cunge et al. 980, Mc Bride and Rutherford, 984), the dispersion of contaminants in estuaries (Winterwerp, 983), and has been solved by researchers like B.J.Noye [4]

2 6 H. Saberi Naafi and H. Hainezhad and others. For the more general transport equation (e. g, with reaction) it has been studied for example by Ataie-Ashtiani et al. [], Khalid Alhumaizi, (Computers & Chemical Engineering. 8(004) ) and Moldrup et al. (994). The aim of this paper is solving the one-dimensional advection-dispersion with reaction, using: FTCS-Ataie, Upstream, DuFort-Frankel, (3-3) and Crank- Nicolson finite difference methods and study the implementation of the methods by comparison the results.. The Mathematical equation The one-dimensional advection-dispersion equation which describes the transient transport of solutes through a homogeneous soil is: t = D C z u kc () z Including first-order reaction, has been taken from [], where C: is solute concentration, (ML 3 ), t: is time, (T ), z: is soil depth, (L), u: is the pore water velocity, (LT ), D: is the dispersion coefficient, (L T ), k: is the first-order reaction rate coefficient, (T ). Since the solute mass leaving from a segment via convection during the time interval t is less than the solute mass inside at the beginning of the time interval. (see [3]), therefore the mass balance criterion of the system is: t u This condition must come with condition stability to show the stability criteria. 3. The finite-difference methods 3.. The FTCS-Ataie method [] The equation () evaluated at the (,n)th grid point, using the FTCS-Ataie, the equation becomes C n+ =( D t () + u t )Cn +( k t D t () )Cn +(D t () u t )Cn + () Where D = D + t (u kd),u = u ( t)ku, k = k t k Ataie et al, in [] investigated the stability of this method using the matrix method and concluded that to satisfy the stability criteria if t D + k

3 Finite-difference methods The Upstream method [4] In this method the backward-difference form uses for the spatial derivative in the term u, forward-difference form for the time derivative and the centered z -difference form for the space derivative, the equation () becomes C n+ C n t = D Cn + Cn +Cn u Cn Cn kc n () This leads to the explicit finite- difference formula C n+ =(s + c)c n +( s c k t)cn + scn + (3) Where s = D t and c = u t () It is easy to show by matrix method [4] and Gerschgorin s circle theorem [] that, (3) is stable if k t < and s + c + k t <. Since k>0 it is sufficient to see t < D () + u + k Note: The equation (3) satisfies the consistency conditions (see []) therefore by Lax theorem [4] in this case with well-posed initial condition the solution which approximates the equation () is convergent The DuFort-Frankel method [4] Another explicit method for solving equation () is the DuFort and Frankel method and We use the following approximation for reaction term C n = (Cn+ + C n ) Using this method the equation () becomes C n+ C n = D Cn + (Cn+ + C n () )+C n u Cn + Cn k Cn+ + C n Therefore C n+ = c +s +s + k t Cn c s s k t +s + k t Cn + + +s + k t Cn (4) Using von Neumann [4] and Miller theorem [4], (4) is stable if G = ((s + k t) c 4s )Sin β +4s (s + k t) Note: The equation (4) satisfies the consistency conditions and with wellposed initial condition the solution which approximates the equation () is convergent.

4 64 H. Saberi Naafi and H. Hainezhad 3.4. The (3-3) method In this method, we use these approximations C = n+ C n t n+ z n+ C n+ z C n+ t = [ Cn Cn = [ Cn + Cn +Cn () C = n+ +C n + Cn+ C n+ ] + Cn+ + Cn+ +C n+ ] () Therefore the equation () at (, n + )th grid point evaluated as t n+ = D C z n+ u z n+ kc n+ or C n+ C n t = D [Cn + Cn + Cn () u [Cn Cn + Cn+ therefore (c + s)c n+ ( + c +s + k t)c n+ + sc+ n+ + Cn+ + C n+ + C n+ ] () C n+ ] k Cn+ + C n = (c + s)c n ( (c +s + k t))cn scn + (5) Using von Neumann method shows that the equation (5) is always stable. If the resulting system of equation solves by Thomas algorithm, it is always stable The Crank-Nicolson method [4] This is an implicit method which uses the same approximations as (3-3) method and z n+ = [Cn + Cn Therefore equation () in this case becomes + Cn+ + C n+ ] (c +s)c n+ +(4+4s +k t)c n+ (s c)c+ n+ =(c +s)c n +(4 4s k t)cn +(s c)cn + (6) The von Neumann method shows that the equation (6) is always stable. If the resulting system of equation solves by Thomas algorithm it is stable if c s.

5 Finite-difference methods Numerical tests To show the implementation and behavior of each method and compare the results obtained from the methods, the analytic solution of the equation () will be used, which is [5] C(z, t) = C 0 [exp((u v)z D z vt )erfc( )+exp( (Dt) (u + v)z D z + vt )erfc( )] (Dt) Where v =(u +4kD) the conditions are C =0, t =0,z >0 C =0, t 0,z C = C 0, t 0,z =0 We assume C 0 = mgl, D =.0cm, u =0.5cmh, k =0., 0.h, =.0cm and =.0h The error function is Error = C Num C Ana where C Num and C Ana respectively are numerical and analytical solution. The results are shown in figures -4.The values of k, t and z used at each test are also shown. 5.Summary and Conclusions The finite difference methods applied to the equation () are i) The FTCS-Ataie, ii) The Upstream, iii)the DuFort-Frankel, iv) The (3-3), v) TheCrank-Nicolson. The results obtained from the methods are plotted in the Figures -4 and describe the following items: a)in the DuFort-Frankel method although the centered-difference formula has been used but as Figures and 3 show the error increases fast, because of the coefficients in (4). b) The FTCS-Ataie works better than Upstream method and (3-3) method, because the approximations in FTCS-Ataie are more accurate, see Figures and 3. c) The Crank-Nicolson method works more accurate than FTCS-Ataie, see Figures and 4. Therefore we suggest using the Crank-Nicolson method for solving the equation ().

6 66 H. Saberi Naafi and H. Hainezhad References [] B.Ataie-Ashtiani, D.A. Lokhington, R.E. Volker, Numerical correction for finite-difference solution of the advection-dispersion equation with reaction, J.Contaminant Hydrology, 996. [] B.N. Datta, Numerical Linear Algebra and Aplications, Brooks/Cole Publishing company, 995. [3] W.Kinzelbach, Groundwater Modeling: An Introduction with Sample Pograms in Basic, Elsevier Amsterdam,986,-65 [4] B.J.Noye, Finite difference Method For Partial Differential Equation, Applied Mathematics Leacture Notes, Department of Applied Mathematics, The university of Adelaide, Australia., 993. [4] M.Th.Van Genuchten and W.J.Alves, Analytical solution of the one-dimention convective-dispersive solute transport equation. U.S. Dep. Agric.,Tech. Bull. No. 66, 98. Received: March 9, 008

7 Finite-difference methods 67 Figure : Shows that errors of the five methods, with k=0., z=0 and t=0()30. The minimum error belongs to the FTCS-Ataie method and Crank- Nicolson method. Figure : shows that errors of Crank-Nicolson and FTCS-Ataie methods with k=0., t=00 and z=0()0. The Crank-Nicolson has the minimum error.

8 68 H. Saberi Naafi and H. Hainezhad Figure 3: Shows that errors of the five methods, with k=0. z=0 and t=0()5. The minimum error belongs to the FTCS-Ataie method and Crank- Nicolson method. Figure 4: shows that errors of Crank-Nicolson and FTCS-Ataie methods with k=0., t=00 and z=0()0. The Crank-Nicolson has the minimum error.