REFORMAS EN LOS ESTADOS EN MATERIA INDÍGENA

Many of the schemes used in this thesis solve the non-conservative form of the advective- diffusion equation. It has been demonstrated that if the concentration profile is smooth and differentiable, these schemes could be used to solve the conservative form of the advective- diffusion equation.

A new analytical solution to a particular form of the spatially varying coefficient advection and advective-diffusion equation has been derived. The advection of a quasi-Gaussian function and step profile were considered. The analytical solution of the spatially varying coefficient advective-diffusion equation for the solution of a step profile was also derived. These analytical solutions can be used to verify numerical models for the solution of the advection and advective- diffusion equations. They are also useful for investigating the behaviour of numerical schemes and for the development and testing of higher-order time-stepping schemes.

Numerical studies have verified the theoretical rates of convergence of traditional finite difference schemes obtained using the modified equivalent partial differential equation technique. The modified equivalent partial differential equation is useful in predicting the accuracy and behaviour of a numerical scheme for the simulation of a smooth profile. However, this is only valid for two time level schemes.

The L u L2 and L* norms for the results obtained from standard finite difference schemes used to solve problems with smooth profiles have identical rates of convergence. The use of shape preserving, slope or flux limiters in a numerical scheme produces different convergence rates for the L u L2 and L«, norms. This was also the case when limiters were used in higher-order time-stepping schemes. The L x norm corresponds to the theoretical rate of convergence of a numerical scheme. The L» norm represents the lower limit of the convergence rate of the numerical scheme. The convergence rate for the L2 norm interpolates the convergence rates for the L x and L« norms. These results highlight the importance of using an appropriate measure of the accuracy of a scheme. The L x norm provides a reliable measure of the convergence rate of a numerical scheme.

The modified equivalent partial differential equation technique assumes that the concentration profile is smooth and continuous. The accuracy of the numerical scheme is significantly reduced if there is a discontinuity in the initial or boundary conditions. Analytical results and numerical experiments suggest that the order of accuracy of conventional finite difference schemes will approximate a discontinuity with 0 (Axö/(p(ß+1)) in the Lp norm sense for all values of p, where

Q is the theoretical order of accuracy of the numerical scheme. Therefore, a sharp front can only be modelled to first-order accuracy at best.

Conclusions and Recommendations 7.2

Models which employ the Laplace transform to solve the advection or advective-diffusion equation have significant advantages over conventional time-stepping schemes. The temporal term in the governing equation is evaluated exactly using the Laplace transform. The effects of the time derivative on accuracy and stability is no longer a consideration. It does not require more computer storage than conventional finite difference or finite element schemes. However, it is only applicable problems which are linear or have been linearized.

Numerical Laplace inversion techniques are required to solve the system of simultaneous equations that result from the application of the Laplace time finite analytic space method. Two well known methods were examined, the Stehfest and the de Hoog et al. algorithms.

The Stehfest algorithm, which operates in the real domain may provide reasonable results for smooth profiles and it is more efficient than the de Hoog et al. algorithm which operates in the complex domain. The Laplace transform inversion algorithms, which operate in the real domain, lose many of their advantages compared to algorithms, which operate in the complex domain, when there are steep gradients in the concentration profile. The Stehfest algorithm is not a suitable technique for inverting the Laplace transform equations in the Laplace time finite analytic space scheme for the solution of the advection and advective-diffusion equations. The de Hoog et al. algorithm is more robust and can resolve discontinuities in a profile accurately. This approach produced results that are far superior to those obtained using any of the other schemes described in this thesis and it is potentially the most efficient method as well.

Currently there is no automatic procedure for locating the poles and selecting the parameters required by the numerical Laplace inversion algorithm which operates in the complex domain. Trial and error is required. This assumes that the form of the solution is known, however in many practical problems this is not the case. Therefore, there is some doubt that the Laplace time finite analytic space scheme will be extensively used to solve practical problems. However, they can be used to validate other numerical models for the solution of the conservative or non­ conservative forms of the spatially varying coefficient advective-diffusion equation for small and large Peclet numbers.

The Quasi-characteristic scheme, which incorporates implicitly the transport properties of the problem and provides the solution on a rectangular grid was described in detail. It only requires spatial interpolation of the known concentrations and it can be used for any flow velocity, including zero-velocity and pure-diffusion cases.

There is no stability restriction imposed on the Quasi-characteristic scheme for the solution of the constant coefficient advection equation. However, linear stability analysis of the Quasi- characteristic scheme for the solution of the advective-diffusion equation indicates that the scheme has a stability criterion which is approximately one-half of that imposed on many two time level explicit finite difference schemes. Boundary information required by the Quasi- characteristic scheme is relatively simple to provide at the required accuracy.

The accuracy of the Quasi-characteristic scheme for the solution of the constant coefficient advection equation depends on the accuracy of the interpolation scheme used. Using a ßth order interpolating scheme will result in a (<2-l)th order time-stepping scheme. The constant coefficient advective-diffusion equation is modelled to first-order accuracy.

were the fourth-order Cubic splines, cubic Hermite, Taut splines, monotone cubic Hermite exponential interpolants, Exponential splines and the sixth-order Quintic splines.

The use of Quintic and Cubic splines in the Quasi-characteristic scheme results in non-physical negative concentrations in the vicinity of abrupt changes in the concentration profile. However, there is excellent resolution of the step and smooth profiles.

The cubic Hermite interpolant did not provide any significant advantages over Cubic splines and it did not possess the shape preserving properties of the monotone cubic Hermite interpolant. The problems encountered with the use of Cubic and Quintic splines were overcome with the use of shape preserving interpolants. These shape preserving strategies could be used with other characteristic based schemes.

Taut splines were found to be unsatisfactory. Continuity of the second derivative is lost when an additional knot is used to preserve convexity of the data. There is a deterioration in the accuracy of the interpolant and estimates of the second derivative which is required for the diffusion term.

The use of monotone preserving cubic Hermite interpolation eliminates the oscillations observed when Cubic or Quintic splines are employed. The strategy used to preserve monotonicity of the data has the effect of introducing diffusion in the numerical scheme. Although not as diffusive as first-order schemes, there is smearing of the step profile and clipping of extrema associated with smooth profiles. This is a problem associated with C 1 interpolants which use local information only to preserve monotonicity of the data.

The problem encountered with the used of monotone preserving cubic Hermite interpolants was overcome with the use of shape preserving C 2 cubic interpolants such as Exponential interpolants and splines. Unfortunately, these interpolation schemes require estimation of tension parameters. Currently there is no analytical procedure for selecting the required tension parameters. Simple strategies for estimating the tension parameters were unsuccessful. A highly successful technique iterates between fitting a monotone cubic Hermite exponential interpolant to estimate the tension parameters and using these parameters to fit a C 2 exponential interpolant to the data. This results in a shape preserving interpolant that eliminates oscillations in the profile and does not clip extrema. Unfortunately the interpolant is not monotone preserving and the iterative procedure required to estimate the tension coefficients makes this approach computationally expensive compared with other schemes used in this thesis. However, the use of shape preserving cubic Hermite exponential interpolants in the Quasi-characteristic scheme is robust, very accurate and universally applicable for the simulation of smooth profiles and profiles with discontinuities for the full range of Peclet numbers.

Obviously, further research is required to develop an efficient method for the selection of tension parameters required by exponential interpolants.

Shape and monotone preserving interpolants were also fitted to the integral of the concentration profile and the concentrations are obtained by differentiating the interpolating function. Fitting a shape preserving interpolant to the smoother integral of the concentration profile is not as computationally expensive as fitting an interpolant to the concentration profile. Unfortunately, it is computational expensive when compared to other schemes.

Conclusions and Recommendations 7.4

Accurate concentration values were not provided by monotone cubic Hermite interpolants and they should not be used. The use of shape preserving Hermite exponential interpolant in the Quasi-characteristic scheme has produced excellent resolution of discontinuous and smooth profiles. This was the case for the solution of the advection and advective-diffusion equations. Although the scheme is only second-order accurate for the solution of the advection equation, it has produced results comparable to third-order schemes. There is an apparent gain in accuracy using the Quasi-characteristic scheme when the interpolant is fitted to the integral of the concentration profile. This result should be verified theoretically.

All schemes based on Quasi-characteristics are only first-order in time for the solution of the advective-diffusion equation. Research into developing higher-order time-stepping is required. The Quasi-characteristic scheme is exact for the solution of the constant coefficient advection equation, otherwise it is only first-order accurate. It is relatively simple to modify the Quasi- characteristic scheme to produce higher order time-stepping. For the solution of the variable coefficient advection equation, high-order time-stepping was obtained using Runge-Kutta integration. Very efficient high-order time-stepping can be achieved using Runge-Kutta integration.

Higher-order time-stepping can also be obtained using modified Richardson extrapolation. The use of the modified Richardson extrapolation will not increase the width of the polluted region surrounding the discontinuity. However, monotonicity is lost and the computational effort required to provide second- and higher-order time-stepping may not be justified. There are alternative schemes such as the efficient ULTIMATE-QUICKEST scheme and Runge-Kutta integration that produces physically realistic results with higher-order time-stepping. The use of the modified Richardson extrapolation is not recommended for practical problems.

The Quasi-characteristic scheme was compared with a number of schemes for the solution of the advection and advective-diffusion equations. The following conclusions can be made regarding the performance of these schemes and their applicability for the solution of these equations. First-order schemes such as the LeVeque and Lax-Friedrichs schemes for the solution of the advection equation and the forward time upwind scheme for the solution of the advection and advective-diffusion equation are not recommended for practical problems. The amount of artificial diffusion introduced by these schemes may be greater than the physical diffusion. This would make it very difficult to calibrate the model to real data.

Second-order finite difference schemes, such as the Lax-Wendroff scheme introduce numerical dispersion. The dispersion results in oscillations in the solution in the vicinity of abrupt changes in the profile.

The width of the region surrounding a discontinuity that is polluted by the first-order and second-order finite difference schemes, combined with their relatively poor accuracy, excludes their use for the simulation of the advection and advective-diffusion equation when steep profiles are involved. However, they are efficient and their use should be restricted to the simulation of profiles that are very smooth.

Filtering techniques have been proposed to eliminate the oscillations that are problematic of second-order schemes. They require special procedures to avoid attenuating extrema. Filtering

techniques perform poorly for both discontinuous and smooth profiles. They are not recommended for practical problems.

Third-order finite difference schemes are capable of producing very accurate results for smooth profiles efficiently. However, near abrupt changes in the profile, the diffusion introduced by the numerical scheme results in oscillations in the solution. They are not recommended for practical problems because negative concentrations are predicted by these schemes.

There is no difference between the Holly and Priessmann scheme and the Quasi-characteristic scheme for the solution of the advection equation. The Holly and Preissmann scheme uses cubic Hermite interpolants and its performance is similar to other third-order schemes. Differences become obvious for the solution of the advective-diffusion equation.

In general, all the second- and higher-order finite difference schemes produce reasonable results for smooth profiles. Some of these schemes produce poor results for profiles with steep regions. All schemes which do not employ some form of slope, flux limiter or shape preserving strategy produced non-physical results when solving problems with steep gradients in the profile. The use of these schemes should be restricted to problems with small Peclet numbers.

The Quasi-characteristic scheme using shape preserving cubic Hermite exponential interpolation produced similar results to the Morrow and Noye and time-splitting schemes for the solution of the advective-diffusion equation. All these schemes have eliminated the oscillations present in the solutions from other second- and higher-order schemes.

The ULTIMATE-QUICKEST scheme is efficient, easy to implement and applicable for the solution of both the conservative and non-conservative advection equations. It performs reasonably well for both smooth and discontinuous profiles and it does not pollute a large region surrounding a discontinuity. However, it is not as accurate as Quasi-characteristics employing cubic Hermite exponential interpolants. But it is significantly more efficient than the Quasi­ characteristic scheme, which is only a first-order time-stepping scheme. It is a third-order scheme in both time and space. It would be worthwhile considering this scheme for large problems or long simulation times. In its present form, Quasi-characteristics with cubic Hermite exponential interpolants, fitted to either the concentration profile or the integral of the concentration profile, would be computationally prohibitive under these circumstances.

The ULTIMATE-QUICKEST scheme can be combined with standard finite difference schemes for the solution of the diffusion scheme using Strang splitting to produce an accurate time­ splitting scheme for the solution of the advective-diffusion equation.

The Cox and Nishikawa TVD scheme is a reasonable scheme for solving the advection of a profile with discontinuities. Although it is only second-order accurate, it exhibits comparable accuracy to other third-order schemes for the simulation of the advection of a discontinuity. This is due to the upwinding compression parameters that are used in the scheme producing a sharp resolution of discontinuities. Morrow and Noye’s third-order flux limiter strategy of Zalesak performs better than the other high-order schemes for this problem. It has produced a slightly better convergence rate than other-third order schemes. The Zalesak flux limiter is more suited to the resolution of a shock than any of the other slope limiter schemes examined. However, its performance is inferior to the ULTIMATE strategy for smooth profiles. The cost of solving two computational schemes in the flux corrected transport scheme of Morrow and Noye is more than

Conclusions and Recommendations 7.6

twice that required for other schemes of comparable accuracy. Both of these schemes are reasonable for the simulation of profiles with discontinuities but not for the simulation of smooth profiles.

It is not possible to make generalizations on the behaviour of numerical schemes based on the width of the polluted region surrounding a discontinuity. All higher-order, slope limiter and TVD

schemes have polluted regions that are at least an order of magnitude smaller than standard finite difference schemes with comparable accuracy. The schemes with robust slope limiter schemes such as QUICKEST, Zalesak’s flux limiter and minimod limiter of Cox and Nishikawa have successfully identified and preserved plateaus in the data. The width of the polluted region surrounding the discontinuity remains constant for schemes which employ these limiters. This was not the case for other schemes.

The ULTIMATE-QUICKEST scheme is recommended for the solution of the advection equation if the width of the region surrounding a discontinuity that is polluted by a numerical scheme is an important criterion in the selection of a numerical model.

The following recommendation can be deduced based on the results obtained in this thesis. For the solution of the advection equation the ULTIMATE-QUICKEST scheme is recommended. Although not as accurate as Quasi-characteristics using shape preserving interpolants, it is computationally efficient and relatively accurate. If computational effort is not a consideration then Quasi-characteristics with shape preserving interpolants fitted to the integral of the concentration profile are preferred.

The flux corrected transport model of Morrow and Noye with the flux limiter of Zalesak is worth considering for the solution of the non-conservative form of the advective-diffusion equation. A time-splitting scheme utilizing the ULTIMATE-QUICKEST scheme is also an alternative scheme for the solution of conservative form of the advective-diffusion equation. This scheme is recommended because it is efficient and produces reasonably accurate results for a wide range of problems. If computational effort is not a consideration, the Quasi-characteristics using shape preserving cubic Hermite interpolation is robust and consistently provides accurate results. It is simple to implement and suitable for a whole range of problems.