As [h.sub.i] [right arrow] 0, the

truncation error tends to zero, which shows that scheme is consistent.

The relative

truncation error is set to be [[epsilon].sub.t] = [10.sup.-3] for the [mathematical expression not reproducible] is used to define the relative error of the H-LU factorization, where [[parallel]x[parallel].sub.F] denotes the Frobenius norm and I is an identity matrix.

So, the local

truncation error for [y.sup.n+1] is O([h.sup.4]).

Now, we will discuss the local

truncation error of the nonstandard scheme (10).

We do not calculate the ROC of the TOMs and BVM8 because their errors are mainly due to round-off errors rather than to

truncation errors. Figure 2 shows the efficiency curves of these methods.

The LUT method will consume excessive memory resources to achieve high accuracy; otherwise, the output signal of the LUT will be affected by

truncation errors due to the limited data depth.

where the term O represents the order of the

truncation error. Similarly, the first numerical partial derivative with respect to [delta] (and all other variables constant) with a second-order

truncation error centered finite difference would be given by

1 that as the

truncation error bounds becomes tighter as the truncation level, N, increases.

To bound the local discretization errors [[member of].sub.n], we need the following

truncation error estimation.

Because the obtained analytical expressions were in the form of infinite summation series, their respective

truncation error bounds are also calculated.

The adaptive estimation methods in [9-11] have been adopted to improve the

truncation error. Instead of exhaustive computing resource simulation methods in previous works [9-11], the QE of SPPC multipliers is analyzed and derived from a simpler statistical method.

Equations (50) and (51) show the local

truncation error of the numerical scheme (24) with respect to the PDE (4).