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To confirm the rates of convergence of conservative schemes the conservative equation

dc d(uaxc)

— + --- = 0

dt dx t > 0, u0 > 0

with the following initial and boundary conditions;

c(x,0) -ln(xlXo) 2

2 a2 and

c(Q,t) = c(oof) = 0 will be solved using the Lax-Wendroff, QUICKEST and ULTIMATE- QUICKEST schemes. These schemes are examples of a naive finite difference scheme, a high resolution scheme and a high resolution scheme incorporating a slope limiter. However, all

Accuracy o f Numerical Schemes fo r Modelling the Advection Equation 5.22

schemes are conservative.

This problem has the following analytical solution (see, equation B.9, Appendix B)

-(InixlXf) - ujf

c(x,t) exp

2 a2

In the hypothetical example, u0 = 0.1, a = 0.2, c0 = 20, x 0 = 0.2 and 0 < x < 2. The

computational time step At = Ax, which corresponds to the Courant numbers o f Cr = 0 at

x = 0 and Cr = 0.2 at x = 2, satisfies the stability constraints imposed on explicit schemes.

The initial conditions in the numerical model correspond to the analytical solution at t = 6.0 and

the analytical and numerical solutions at t = 10 were compared.

The analytical solution at times t = 6 and 10 is shown in Figure 5.16, as well as the predicted

results for the Lax-W endroff scheme using 201 evenly spaced computational nodes in the interval 0 < x < 2. Ö 80. t - 10 o 40. .0 .5 1.0 1.5 2.0 d i s t a n c e

Figure 5.16 Analytical and Lax-W endroff scheme solution o f the conservative advection equation of a smooth profile in a non-uniform velocity field

The L u L 2 and L«, norms were calculated for the deviation between the numerical and analytical

profiles at t — 10 using all the computational nodes. The computational domain was discretized

by fifteen data sets containing between 201 and 20001, (Ax = 0.01 to 0.0001), computational

nodes.

The theoretical rate of convergence o f the Lax-W endroff scheme is (^(A/^Ax2). The convergence

rate o f the QUICKEST scheme is 0(At3,Ax}). Using the numerical study, the rate o f convergence

o f the Lj norm for the Lax-W endroff and QUICKEST schemes are shown in Figure 5.17. The

A QUICKEST

- 1 0.

-4 .0

L o g ( A x )

Figure 5.17 Rate of convergence of the Lax-Wendroff and QUICKEST schemes for the solution of the conservative advection equation with variable coefficients

□ L

A A

A L „

- 4 .0 -3 .0

L o g ( A x )

Figure 5.18 Rate of convergence of the ULTIMATE-QUICKEST scheme for the solution of the conservative advection equation with variable coefficients

Accuracy o f Numerical Schemes for Modelling the Advection Equation 5.24

is third-order accurate. Similar convergence rates were obtained for the and L x norms. The convergence rate for the L u L2 and L x norms for the Leonard ULTIMATE-QUICKEST

scheme are shown in Figure 5.18. With the exception of the Lx norm, which has third-order rate of convergence, the remaining norms show a reduction in the rate of convergence, with the L«, norm behaving like 0{Ax2). This is consistent with the results shown in Figure 5.3. The

ULTIMATE-QUICKEST scheme is third-order accurate for the L x norm only. Slightly lower convergence rates were obtained for the L^ and L«, norms. The deterioration in the convergence rates in the ULTIMATE-QUICKEST scheme, for both the uniform and non-uniform velocity field, is attributable to the poor performance of the ULTIMATE strategy at extrema.

5 .3 .2 Non-Conservative Schemes

The application of higher-order time-stepping will be illustrated using the first-order in time Holly and Preissmann scheme for the solution of the following problem.

— + ujc— = 0 r > 0, > 0

dt ^ dx 0

with the following initial and boundary conditions;

c(x,0) ---c0 exp

o\j2r

- ln(x/xo)2 2a 2

c(Of) = 0 and c(oo,t) = 0. This is the non-conservative spatially variable coefficient form of the advection equation.

In this hypothetical example, u0 = 0.1, a — 0.2, c0 = 20, x 0 = 0.2 and 0 < x < 2, which are identical to the values used in the above example. The computational time step used is At = Ax.

The following analytical solution was derived for this problem in Appendix B, (equation B.10)

c(xf) ---C0 exp

ct\/2t

-■(ln(x/xq) - ty )2 2 o2

The initial conditions in the numerical model correspond to the analytical solution at / = 6.0 and the analytical and numerical solutions at t = 10 were compared. The analytical solution at times

t = 6 and 10 are shown in Figure 5.19, as well as the predicted results from the Holly and Preissmann scheme.

The characteristics for the advection equation are curved in a non-uniform velocity field. The characteristics are straight in uniform velocity fields and the accuracy of the time-stepping is irrelevant. Any two time level scheme can be employed. Time-stepping may be viewed as integration along these characteristics. The use of first-order time-stepping or simple averaging in time is sufficient to evaluate the time-stepping exactly for constant coefficient problems. However for curved characteristics they are only first-order approximations. This can be

t - 1 0

t = 6

£ 6 0.

d is t a n c e

Figure 5.19 Analytical and Holly and Preissmann scheme solution of the non-conservative advection equation for a profile in a non-uniform velocity field

O Without Runge-Kutta

□ With fourth-order Runge-Kutta

-3 .5 -3 .0 -2 .5 -2.0

Figure 5.20 Convergence rate of the Holly and Preissmann scheme solution of the variable coefficient non-conservative advection equation with and without

Accuracy o f Numerical Schemes for Modelling the Advection Equation 5.26

confirmed using the Holly and Preissmann scheme to solve the above problem.

The L u L2 and L „ norms were calculated for the deviation between the numerical and analytical profiles at t — 10 using all the computational nodes. The computational domain was discretized by fifteen data sets, containing between 201 and 20001 computational nodes, (Ax = 0.01 to 0.0001). The rate of convergence of the Holly and Preissmann scheme for the non-uniform velocity field problem is shown in Figure 5.20. Although the scheme is third-order accurate in space, it is only first-order accurate in time.

High-order time-stepping is required to obtain higher accuracy in time for unsteady or non- uniform flow problems.

A number of schemes are available to produce higher-order time-stepping. These include; (i)

developing finite difference schemes which span a number of time steps, (ii) using the method o f lines to reduce the problem to solving an ordinary differential equation in time to the desired accuracy and (iii) extrapolation techniques.

(a) Multi-time Level Schemes

Noye (1991) describes a number of three time level finite difference methods for simulating advection in one dimension. Noye (1992) developed a Leapfrog method for solving the one­ dimensional advection equation in a subsequent paper. Although three level schemes will generally produce greater accuracy than two level schemes for smooth initial boundary data, there is only a slight increase in accuracy when the initial boundary data is discontinuous (see, for example Noye [1991]). In addition, there is a problem of providing the two time levels of initial values required by three time level schemes. One option is to use a two time level scheme of comparable accuracy to provide these values.

The disadvantages of multi-level type methods are; (/) for an m-level method the first m- 1 levels have to be calculated by other methods to the same order of accuracy, therefore they are difficult to start-up and (ii) m levels of data must be stored.

(b) Method of Lines

Discretizing the spatial term first reduces the partial differential equation into an ordinary differential equation in time of the form

where Ca is a linear algebraic relationship between the nodal concentrations. This process of reducing partial differential equations into ordinary differential equations is an example of the

method o f lines (see, for example Shu and Osher [1988] and Fletcher [1991]). For conservative finite difference schemes

Ax

Any spatial discretization scheme, such as those described above for conservative schemes, can be used to evaluate the flux terms.

This is an attractive alternative to the application of simple finite differences for the time derivative because the large amount of knowledge that is available for the solution of ordinary differential equations can be utilized.

There are a number of techniques available for the solution of equation (5.6) (see, for example Rice [1983] and Stoer and Bulirsch [1992]). These include in order of sophistication; (/) Euler, (ii) Runge-Kutta, (iii) multi-step and (iv) predictor-corrector.

The Euler method is simply

The two time level explicit schemes described in Section 2.3.1.2, such as the forward time centred space, forward time upwind and Lax Friedrichs schemes, could be considered as Euler schemes. These are only first-order accurate in time.

Predictor-corrector methods use previously stored solutions to extrapolate the solution one step in advance. A correction is applied to the extrapolated solution using derivative information at the new point. Iterating with the corrector is required to obtain the desired accuracy. In practice this may only involve two or three iterations.

Because derivative information is required, predictor-corrector methods are best suited for very smooth functions. Predictor-corrector and multi-step methods require large storage areas to store the concentration vectors over a number of time steps and they are difficult to start-up. Fletcher (1991, p. 243) suggests that higher-order temporal schemes, such as multi-step and predictor- corrector schemes are not warranted because errors in the solution would be dominated by the spatial discretization errors. Therefore, the relatively low storage Runge-Kutta schemes are preferred.

The solution in Runge-Kutta methods is obtained using successive Euler-type steps. The information from these steps is assembled so that the resultant scheme matches a Taylor series expansion of the solution to the desired accuracy.

Runge-Kutta type methods have the advantage of being self-starting and requiring smaller storage compared to multi-stepping and predictor-corrector methods. Multi-level methods may be simpler for very high-order methods (> 5 ) (Shu [1988]).

Cjn + A tC (Cjn).

Accuracy o f Numerical Schemes for Modelling the Advection Equation 5.28

cn+1 = cn + At Y , a a Ca(cn+i/m) i = 1 ,2 ,...,m (5 -7)

*=o

in which otik is a constant to be determined by requiring that cn+1 satisfies the Taylor series

dc T t

;n+1 = c n + Ar— n 4- Ar2 d2c n 4 - At3 d3c