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Verification of the discretization--order investigation

In order to verify the foundation of the neutronics program block, the discretization procedure which leads to the calculation of the matrices $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ and $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ in the eigenvalue problem, it is customary to verify the order of the numerical method.

At this stage it is necessary to introduce some terms. The numerical solution to the continuous space multi-group diffusion equation, (1.16) (see page ), denoted by $\{ {{\lambda_0}}, {\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}} \}$ is obtained using the following procedures, which all introduce error.

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Discretization of the originally continuous space eigenvalue problem.

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Calculation of the different parts of the discretized eigenvalue problem (the two block matrices).

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Iterative solution of the discretized eigenvalue problem--including the solution of a block tri-diagonal system of equations.

If we call the exact solution to the continuous space eigenvalue problem (1.16) $\{ \hat{\lambda_0} , \hat{\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}} \}$ we define the following errors. The (total) error on the eigenvalue, $E_{\lambda}$, is simply defined as

$$E_\lambda \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscriptstyle\triangle$}}\;\lambda_0 - \hat{\lambda_0}$$ (4.19)

The error on the eigenvector is, however, for several reasons more difficult to define. It is in general a bad idea merely to subtract the exact eigenvector from the numerically calculated one and the investigate the individual components of this vector. This is so because an eigenvector specifies a direction in the Nth dimensional space and one is therefore bound to investigate the difference in the direction in stead of the difference of individual vector components. A suitable measure of the directional error on the numerically computed eigenvector, which come to mind is the angle between the two vectors. In mathematical terms we define the (total) error on the eigenvector, $E_{\phi}$, as

$$E_\phi \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptsc... ...0.15ex},\hat{\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}})$$ (4.20)

The suitability of this error measure is confirmed later in this section by experiment.

If the steplengths in the grid are not too small (such that round off errors can be neglected) and if we apply a very strict convergence criterion (eg 10^-14) when we solve the discretized eigenvalue problem using the power method and/or the fractional iteration method we can assume that the (total) errors (4.21) and (4.22) are dominated by the discretization error which arises in connection with the discretization of the continuous space eigenvalue problem. When this is the case we in most cases have the following behavior of the (total) errors ([13])

$$E(h) = C_1 h^q + C_2 h^{q+1} + \cdots \qquad\qquad, \quad C_i,q \in \Re$$ (4.21)

where h is the steplengths in the computational grid and q at best corresponds to the order of the principal truncation error (see (1.45) p. ) of the discretization. That is, if all the steplengths in the computational grid are reduced by a factor of two, the error is reduced by a factor of 2^q.

If we assume that the error has the form (4.23) and that the steplength is so small that the higher order terms become small, we can estimate the order of the error, q, in the following way

$$q \approx \frac{1}{\log_{10}(2)} \log_{10}\left[ \frac{ E(4h) - E(2h) }{ E(2h) - E(h) }\right]$$ (4.22)

It is evident that this expression, at least in the case of the eigenvalue (see (4.21)), does not require knowledge of the exact value (denoted by a $\;\hat{\rule{0mm}{0.7em}\;\;\;}\;$). The same cannot be said about the eigenvector, this would require a thorough mathematical investigation of theoretical nature which we will not go into. We simply state that acceptable results are obtained by replacing the errors in (4.24) with the angles between successive numerically calculated eigenvectors, ie E(h) is for instance replaced by $\angle (\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}(h),\hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}(2h))$.

In regard to choosing an appropriate test reactor it was felt that a real power reactor would test the numerical method more thoroughly because the cross sections vary throughout the core of the reactor. The test reactor used in this section consequently corresponds to a reactor which consists of $(8\times8)$-GE BWR assemblies (without gadolinium burnable poison). The cross sections vary throughout the core as a result of a linearly varying fuel temperature (T_f) and void fraction ( $\mbox{$<\!{\alpha}\!>$}$).

We start out with a computational grid specified with the steplength vector $\hspace{0.2ex}\underline{h}{}\hspace{0.15ex}_0$ which has steplengths of $5 \mbox{cm}$ in core regions and $1 \mbox{cm}$ in reflector regions and we use 10 transition points (Growth factors: 1.17 and 0.85). We then calculate the eigensolution for successive reductions of the steplengths in the grid--the results of the computations are seen in Table 4.2. The last item in the table is the infinity norm of the residual vector for the computed eigensolution, $\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}$, defined by (Cf. (1.91))

$$\hspace{0.2ex}\underline{r}{}\hspace{0.15ex} \;\hbox{$=$\ke... ...hspace{0.15ex} \hspace{0.2ex}\underline{\phi}{}\hspace{0.15ex}$$ (4.23)

If we apply a very strict convergence criterion on the eigensolution obtained by the fractional iteration method, $\Vert\hspace{0.2ex}\underline{r}{}\hspace{0.15ex}\Vert _\infty$ tells us how pronounced rounding errors are.

It is unexpected that we obtain a 2nd order behaviour of the applied difference scheme since we in section 1.5 saw that the scheme in this case should only have first order accuracy. The likely reason for this phenomenon is according to [13] that the distribution of local truncation errors at grid points after a number of grid reductions show the pattern^4.1 displayed in Figure 4.20. As we successively reduce the steplengths by factors of 2 the number of points with first order accuracy ${\cal O}(h)$ is fixed while the number of points with second order accuracy ${\cal O}(h^2)$ grows and this phenomenon can in certain circumstances result in a second order behaviour of the discretization error, according to [13].

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