An operator splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations

Abstract An operator splitting algorithm for a system of one-dimensional advection-diffusion-reaction equations, describing the transport of non-conservative pollutants, is presented in this paper. The algorithm is a Strang type splitting procedure incorporating contributions from the inhomogeneous terms by the Duhamel's principle. The associated homogeneous equations are split into advection, diffusion and reaction equations, and solved by a backward method of characteristic, a finite-element method and an explicit Runge-Kutta method, respectively. The boundary conditions applicable to the split equations are derived. Numerical analyses of the algorithm, consisting of the stability, the accuracy and the convergence of the solution procedure, are presented. The composite algorithm is second-order accurate in time and space and conditionally stable. The numerical characteristics of the algorithm are demonstrated by several examples.

[1]  K. Higashi,et al.  Analytical Models for Migration of Radionuclides in Geologic Sorbing Media , 1980 .

[2]  M. Th. van Genuchten,et al.  Convective-dispersive transport of solutes involved in sequential first-order decay reactions , 1985 .

[3]  Cass T. Miller,et al.  Development of split‐operator, Petrov‐Galerkin Methods to simulate transport and diffusion problems , 1993 .

[4]  F. Holly,et al.  Accurate Calculation of Transport in Two Dimensions , 1977 .

[5]  P. Belleudy,et al.  A higher‐order eulerian scheme for coupled advection‐diffusion transport , 1991 .

[6]  An operator‐splitting algorithm for two‐dimensional convection–dispersion–reaction problems , 1989 .

[7]  G. Pinder,et al.  Numerical solution of partial differential equations in science and engineering , 1982 .

[8]  J. Bear,et al.  Modeling groundwater flow and pollution , 1987 .

[9]  C. Cho,et al.  CONVECTIVE TRANSPORT OF AMMONIUM WITH NITRIFICATION IN SOIL , 1971 .

[10]  Leszek Demkowicz,et al.  A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting method and h-p adaptivity , 1990 .

[11]  Randall J. LeVeque,et al.  Numerical methods based on additive splittings for hyperbolic partial differential equations , 1981 .

[12]  Wolfgang Kinzelbach,et al.  Numerical Modeling of Natural and Enhanced Denitrification Processes in Aquifers , 1991 .

[13]  A. Gourlay,et al.  Intermediate boundary corrections for split operator methods in three dimensions , 1967 .

[14]  Martinus Th. Van Genuchten,et al.  Analytical solutions for chemical transport with simultaneous adsorption, zero-order production and first-order decay , 1981 .

[15]  W. G. Gray,et al.  An analysis of the numerical solution of the transport equation , 1976 .

[16]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[17]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[18]  Investigation of use of reach‐back characteristics method for 2D dispersion equation , 1991 .

[19]  Albert J. Valocchi,et al.  Accuracy of operator splitting for advection‐dispersion‐reaction problems , 1992 .

[20]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[21]  Hanadi S. Rifai,et al.  Comparison of biodegradation kinetics with an instantaneous reaction model for groundwater , 1990 .

[22]  J. Tinsley Oden,et al.  Finite Elements, Mathematical Aspects. , 1986 .

[23]  T. N. Narasimhan,et al.  A multiple species transport model with sequential decay chain interactions in heterogeneous subsurface environments , 1993 .

[24]  R. LeVeque.  Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations , 1986 .

[25]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[26]  P. Mccarty,et al.  Chemistry for environmental engineering , 1978 .

[27]  Philip J. Rasch,et al.  On Shape-Preserving Interpolation and Semi-Lagrangian Transport , 1990, SIAM J. Sci. Comput..

[28]  Mary F. Wheeler,et al.  An Operator-Splitting Method for Advection-Diffusion-Reaction Problems , 1987 .

[29]  On the treatment of time‐dependent boundary conditions in splitting methods for parabolic differential equations , 1981 .

[30]  G. Marchuk.  Splitting and alternating direction methods , 1990 .

[31]  F. Holly,et al.  Dispersion Simulation in Two‐dimensional Tidal Flow , 1984 .

[32]  C. Dawson,et al.  Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport , 1992 .

[33]  W. Rodi,et al.  A higher order numerical scheme for scalar transport , 1982 .

[34]  J. B. Perot,et al.  An analysis of the fractional step method , 1993 .