The one-dimensional non-conservative advective-diffusion equation with a reaction and sink- source term considered here is
3c(x’f) + u(x)dc(xJ) = D(x) - K(x)c(x,t) + S(x,t) (4.2)
dt dx dx2
in which the velocity and diffusion coefficients are considered as functions of distance. Taking the Laplace transformation of both sides of (4.2) gives
sC(x,s) - c(x,0) + u(x)dc(x’s) = D(x)älc{x's) - K(x)c(x,s) + S(x,s). (4.3)
dx dx2
For convenience, the reaction and sink-source terms will be ignored and equation (4.2) is subject to the following initial and boundary conditions c(x,0) = 0, c(0,r) = c0 and c(°° ,t) = 0. Equation (4.3) then becomes
sC(x,s) + u(x)dc(-x ’sl = (4.4)
dx
and the transformed boundary condition is
5(0,J) = L
The temporal terms in equation (4.2) have been evaluated exactly using the Laplace transform. The effects of the time derivative on accuracy and stability is no longer a consideration. The Courant number, Cr — uAt/Ax, is no longer a constraint. The accuracy of the scheme now only depends on the method used to solve equation (4.4) and the technique used to invert the resultant Laplace space equation.
4.2.1 Solution of the Laplace Space Equation
The steady state transformed equation, equation (4.4), is a second-order ordinary differential equation. This equation can be solved; (/) analytically or (ii) numerically. The choice depends
Quasi-Analytical Method fo r the Solution o f the Advection
and Advective-Diffusion Equations Using Laplace Transforms 4.3
on the complexity of the coefficients in equation (4.2).
For constant coefficients the analytical solution to equation (4.4) is (see, Appendix A)
UX X u2 m'
2 0
---- + s
4 D
(4.5)
This can be inverted analytically to obtain the following exact solution to equation (4.2).
c(x,t) = - 1 erfc x - ut + — coexp ux erfc x + ut
2slDt 2 D 2 s/Dt
Although a useful solution, in practice the velocity field and the diffusion coefficient are not constant.
For spatially variable coefficients a number of techniques have been proposed for solving equation (4.4). These include; (i) finite difference of finite element schemes, (ii) finite analytic space method and (iii) analytically.
4.2.1.1 Laplace Time Finite Difference or Finite Element Space Methods
Equation (4.4) can be solved using standard finite difference of finite element techniques. Moridis et al. (1991) used a centred finite difference scheme for the spatial discretization and the Laplace transform for the time stepping. Sudicky (1989) employed a finite element scheme for the spatial approximation. Li et al. (1992) found that for small Peclet numbers, Pe = uAx/D,
the Laplace time centered finite difference space method was very accurate. However, for Peclet numbers of 20, 50 and 500, oscillatory solutions are observed. These oscillations were more pronounced than those observed in implicit time centered space finite difference schemes. This was explained by the fact that the temporal discretization in the implicit scheme introduces diffusion, which dampens the oscillations. The numerical diffusion does not exist in the Laplace method where the temporal discretization is exact.
It seems that many of the schemes which use finite differences or finite elements to approximate the spatial derivatives exhibit the same characteristics as those methods which employ finite differences to approximate both the spatial and temporal derivatives. They introduce either artificial diffusion or dispersion in the solution (see, for example Li et al. [1992])
4 .2 .1 .2 Laplace Time Finite Analytic Space Method
The Laplace time finite analytic space method is described in detail by Chen et al. [1981]. In this method, the spatial domain 0 < x < L is discretized into A subdivisions, 0 = x0 < x u ... ,
xjA < Xj < xj+l, ..., xNA < xN = L. The local element shown in Figure 4.1, associated with*;, spans three successive computational nodes, xjA, xjy and xj+l.
For each element the coefficients in equation (4.3) can be approximated locally by the constant values
local element
__________ A
r
• —
xj+l
Figure 4.1 Local element in the Laplace time finite analytic space solution o f the advective- diffusion equation
u{x) * Uj, c(x,0) « f jt D(x) ® D , 5 ( ^ , 5 ) » S jt K(x)
where Uj, fj} Dj, Kj and 5. are evaluated at x = Xj. When the original coefficients vary over space, the discrete values at Xj are assigned to the local element so that the Laplace transformed advective-diffusion equation becomes a locally constant coefficient differential equation. In this case equation (4.3) is a second-order homogeneous ordinary differential equation with constant coefficients.
5j + f j (4.6) For each element the following analytic solution
£(x,s) = Axexp[rx(x - Xj)/Ax] + A2exp[r2(x - Xj)IAxj\ + —— ■
Kj +
applies over each element, in which xjA < x < xj+l, AXj = x2 - xjA, and rx and r2 are given by
u . A x.. . . . and 1 + " 1 + 4 Dj(KJ J + s)~ 1/2 “1 r2 uAc. r r 1/2 1 - 1 + 4 J 1D ( K + UJ
The two integration constants A x and A2t can be determined from the boundary conditions
= CJ-V C(* .4 ) l *7+1*
Substituting into (4.6) and solving fo r/t, and A2 yields
exp(r25j)Ch ,- exp(-r2)Cjtl exp(-r2) Sj
exp(r2&j - r{) - exp(rfi - r2) exp(r25j - r,) - exp(rA - r2) A* + s and
-exp(rfi)Cj, I + expt-rJC^ exp(r25j - r,) - exp(rfij - r2)
expjrfij) - exp(-rt) Sj + f: exp(r2dj - r{) - exp{rxbj - r2) A + s
Quasi-Anatylical Method fo r the Solution o f the Advection
and Advective-Diffusion Equations Using Laplace Transforms 4.5
in which 5, = Axj+l/Axjm
Writing equation (4.6) for an interior node, located at xjt produces the following simple algebraic equation
CJ
= “Ä-l + 0Ä.1 +
(4’7)
where the coefficients ajy ßjy and 7 are given by
exp(r2b) - exp(rxb)
OL- = --- --- --- ,
; exp{r2bj - r{) - exp{rfi. - r2)
ß . = ______expS : r ^ r ____ and
; exp{r2bj - rx) - exp{rxbj - r2)
exp(rxb) - exp(r2b) - exp(-rv) + exp{-r2) S + f
y . = 1 + ---1---1---1---i. J exp(r2bj - rx) - exp(rfi. - r2) Kj + s
Equation (4.7) provides a linear relationship between the concentration c(;c,s) at the interior node
Xj with the concentrations C(Xj_v s) and d(A:;+1,5) at the two neighboring nodes. These relationships are commonly encountered in finite difference schemes.
Written for all nodes,,/ = 1,2, ..., N -1, equation (4.7) produces a system o f linear algebraic equations with Af + 1 unknown nodal concentrations, Cjy j = 0,1, ... ,N. The boundary conditions £(0,s) and £(L,s) provide the necessary information to solve the system o f equations. The system o f equations can be written in matrix form
[A] {C} = {/?} <4 -8>
where [A] is the coefficient matrix, {R} is the known right hand side vector and {C} is the vector o f unknowns. This defines the desired Laplace time finite analytic scheme. Both [A] and
{R} depend on the Laplace space variable s.
When the coefficients of equation (4.2) are constant the Laplace time finite analytic space solution, equation (4.6), is identical to the exact solution given by equation (4.5), with the omission o f the reaction and source-sink term. For variable coefficients, the Laplace time finite analytic space scheme remains accurate provided that the spatial grid adequately defines the spatial variation in the velocity field and the dispersion coefficient.
The major advantages o f the Laplace time finite analytic space scheme are; (/) there is no need to obtain an analytical or series solution to equation (4.3), (ii) it does not require more computer storage than conventional finite difference or finite element schemes, (iii) it is suitable for non- uniform grid spacing, (z'v) it has an unrestricted time step and (v) unlike the analytical solution, any initial conditions can be imposed on the problem. These are important improvements over conventional time-stepping approaches. Its major disadvantages are that; (/) a system of
equations must be solved and (ii) discretization errors are introduced for variable coefficient problems.
4.2 .1 .3 Laplace Time Analytic Space Method
The finite discretization described above can be eliminated if the variation of the coefficients in equation (4.2) can be incorporated in this equation explicitly. Polynomials, piecewise interpolants and splines could be used to define the variation of the coefficients in space. In this way it is possible to avoid the solution of a system of simultaneous equations in the Laplace time finite analytical method.
The only difficulty is obtaining a solution to the second-order ordinary differential equation with variable coefficients. There are known analytical solutions for a number of polynomial expressions used to define the spatial variation in the coefficients. These include linear Legendre and Euler equations (see, for example Murphy [I960]). Alternatively, it is possible to obtain a series solution to the ordinary differential equation (see, for example Rainville and Bedient [1989]). The use of ‘symbolic manipulation’ computer packages could be of use in finding such analytical solutions.
Second-order ordinary differential equations with variable coefficients, such as equation (4.4), can be reduced into a second-order ordinary differential equation with constant coefficients by introducing a new variable. It is much simpler to obtain an analytical solution to a constant coefficient equation than for a variable coefficient equation. Hildebrand (1962) refers to this class of equations as equidimensional linear equations.