Gorka Moreno

The Laplace transform can be inverted in the complex domain using (Crump [1976])

c(x,t) exp(vt) {Re[£{x,s)\Cos{ut) - Im[€(x,s)]Sin(ut)} du (4.17)

Quasi-Analytical Method fo r the Solution o f the Advection

and Advective-Diffusion liquations Using Laplace Transforms 4.13

r 60.

.0 5. 10. 15. 20.

t

Figure 4.4 Approximation of In(x,t) to c(x,t) = Tanh{5(10 - t)) 4- 1 for various values of n

n = 5 10

The following equation, obtained by discretizing (4.17) using a trapezoidal rule in the interval [0,27], is the basis of many existing Fourier based schemes for inverting the Laplace transform (Sudicky [1989]) c(x,t) = Z Z & L

N.

- E

where E is the discretization error, given by

oo

E(u,t,T) = -^ 2 exp{-2kvT) c(x,2kT+t)

* = i

is introduced because the infinite series in equation (4.17) has been truncated to Ns terms and the step size is tt/T.

The parameters v and T strongly affect the discretization error, E. A large value of T reduces the discretization error. The variable v is the distance from a pole or singularity to where the inverse is to be evaluated, |y| > 0 . The most accurate solution, requiring fewer function evaluations, Ns, is obtained when the solution is in the vicinity of a pole. However, if the solution is sought too close to a pole the inversion may become unstable. Currently there is no automatic procedure for the selection of v, Ns and T. Their selection is an art.

A major limitation to the practical application of numerical schemes that are based on the Laplace transform and its numerical inversion in the complex domain, is that it is often difficult to locate the poles of the Laplace transform equation. The location of the poles are not required for the Stehfest algorithm which operates in the real domain.

Unfortunately, there is no automatic procedure for locating these poles. In Appendix C the poles are located for very simple problems for the Laplace time finite difference space scheme, for the solution of both the advection and advective-diffusion equation. A number of finite difference approximations have been considered and a stability analysis is also presented for these schemes. However, the stability analysis is a linear analysis and assumes that the coefficients are constant. Linear stability analysis and the location of the poles for the Laplace time finite analytic space scheme described above, is not appropriate. However, the Laplace time finite analytic space scheme is exact and unconditionally stable for the constant coefficient problem. A method for locating poles in this case and for variable coefficient problems has not been found.

The Laplace space variable

Quasi-Analytical Method fo r the Solution o f the Advection

and Advective-Diffusion Equations Using Laplace Transforms 4.15

ln(E)

and Crump (1976) found that T = 0.8fmax, E ~ 10'6 yields accurate results for a wide range of times, t, where t E [ 0 , / ^ and tmx is the maximum simulation time. The only remaining variable required is the number of terms, Ns, used in the truncated series, equation (4.18). The summation kernels in these truncated series may be highly oscillatory. One common problem associated with the direct evaluation of equation (4.18) lies in the extremely slow convergence of the series expression with adverse implications for both accuracy and computation time. The value Ns required for a typical problem is often of the order of 103. A number of algorithms have been developed in an effort to accelerate the convergence of equation (4.18). These include; (i) the Euler transform (see, for example National Physical Laboratory [1961], p. 124), (ii) the epsilon algorithm (see, for example McDonald [1964]) and (iii) the

quotient-difference algorithm (see, for example Press et al. [1986], p. 136).

Crump (1976) found the epsilon algorithm to be superior to the Euler transform in speeding convergence of the series in equation (4.18).

The epsilon algorithm applied to a partial sum of a series is equivalent to constructing successive rational approximations to the power series (Wynn [1956]). These rational approximations are Pade approximations to the power series (McDonald [1964]). The Pade approximations generally have a larger region of convergence than the original series. Unfortunately, in the epsilon algorithm, the Pade approximations must be re-calculated for each value of t.

Alternatively, the quotient-difference algorithm produces the rational approximations in the form of a continued fractions. This form facilitates the evaluation of the rational approximation for multiple values of t more efficiently than the epsilon algorithm. De Hoog et al. (1982) improved the convergence rate of the quotient difference algorithm by applying an acceleration procedure to the continued fractions as well. The consequence of the improved acceleration procedure is an increase in accuracy and stability of the inversion procedure. It is a significant improvement over the Crump algorithm. It is more efficient when inversions for multiple values of t are required, produces better resolution of discontinuous functions and is less susceptible to roundoff errors. The value of Ns needed for adequate accuracy is dramatically reduced to approximately 5 to 40 for most problems.

Irrespective of the Laplace inverse technique chosen, since the matrices [A] and {R} in equation (4.8) are dependant on the Laplace space variable, s, a set of Ns vectors

{C}„ = [A];l{R}n n = 1,2 ,...,A

are needed to obtain a solution at time t. Therefore, the system of simultaneous equations has to be solved at least Ns times.

Since the coefficient matrix in equation (4.8) is tridiagonal, the efficient Thomas algorithm can be used provided that the coefficient matrix is diagonally dominant.

For diagonal dominance

Iexp(r2dj - r,) - exp(rxb. - r2)\ > \exp{r2b.) - exp{rxb)\ + \exp(-rl) - exp(-r2)\.

For practical problems K(x) > 0, then rx > 1 and r2 < 0 and the above inequality becomes I - exp(rxSj - r2)| > | - ,)| + | -

which is always satisfied. Therefore, equation (4.8) can be solved using the Thomas algorithm. The number of vectors, Ns required is generally much smaller than the number of time steps required in standard time marching schemes. Therefore, the Laplace transform approach is potentially more efficient, especially for long time simulations, than time-stepping schemes. The Stehfest and de Hoog et al. Laplace inverse algorithms were used to perform the Laplace inversion in the Laplace time finite analytic space simulation of the advection of the test profile. The results of the simulation are shown in Figure 4.6 which was produced using t^ = 100, 7 = 0.1 and Ns = 29 in the de Hoog et al. algorithm and for the Stehfest results Ns = 25.

de Hoog et a t Stehfest