Environmental Engineering Reference
In-Depth Information
Only the first-order derivatives are used. With only the first-order
derivatives, for rapidly changing the step size must be kept suffi-
ciently small in order to meet the stability and accuracy requirements.
Only the information of the present and that of the most recent past
data point are used in estimation of Both the accuracy and
speed can be improved if more past data are used to predict
It is evident that in order to increase the step size and in the mean
time to improve the accuracy, more past data points and higher-order
derivatives should be used in prediction of Calculation of high-
order derivatives numerically is generally not only difficult and but also
costly. For this reason, in practice, multiple past data points are used,
but only the first-order derivatives at these data points are usually used
as a compromise to estimate leading to the linear multi-step
predictor-corrector (LMS-PC) algorithms.
Let a total of past data points, denoted by
and are known. Also assume the first-order time derivatives at these
past data points, denoted by and are known as
well. The predictor of LMS-PC that predicts the next data point
is given by
Note that the predictor uses only the known information of the past
points. The corrector of LMS-PC algorithm is given by
The corrector contains and that are both unknown. Eq. (4.12)
needs to be solved iteratively using Newton-Raphson iterations. Substi-
tuting (4.12) into (4.1)
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