Transport: Non-diffusive, flux conservative initial value problems and ...

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by a truncated Taylor’s series. For example a second order quadratic polynomial
f(x) = ax 2 + bx + c can be described exactly with a Taylor series that contains
only up to second derivatives (all third derivatives and higher are zero). What does
this buy us? Fortunately, we also know (thanks to M. Lagrange) that given any N
points y1 = f(x1),y2 = f(x2),... ,yN = f(xN), there is a unique polynomial of
order N − 1 that passes exactly through those points, i.e.
P(x) = (x − x2)(x − x3)...(x − xN)
(x1 − x2)(x1 − x3)... (x1 − xN) y1+
(x − x1)(x − x3)...(x − xN)
(x2 − x1)(x2 − x3)... (x2 − xN) y2 + ...
+ (x − x1)(x − x2)... (x−xN−1)
(xN − x1)(xN − x2)...(xN − xN−1) yN
(5.3.9)
which for any value of x gives the polynomial interpolation that is simply a weighting
of the value of the functions at the N nodes y1...N . Inspection of Eq. (5.3.9)
also shows that P(x) is exactly yi at x = xi. P(x) is the interpolating polynomial,
however, given P(x), all of its derivatives are also easily derived for any x between
x1 and xN 1 These derivatives however will also be exact weightings of the known
values at the nodes. Thus up to N − 1-th order differences at any point in the
interval can be immediately determined.
f(x)
3.0
2.5
2.0
1.5
1.0
0.0 0.5 1.0
x
1.5 2.0
Figure 5.2: The second order interpolating polynomial that passes through the three points
(0∆x, 1), (1∆x, 2.5), (2∆x, 2)
As an example, Fig. 5.2 shows the 2nd order interpolating polynomial that
goes through the three equally spaced points (0∆x,1), (1∆x,2.5), (2∆x,2) where
1 The derivatives and the polynomial are actually defined for all x however, while interpolation is
stable, extrapolation beyond the bounds of the known points usually highly inaccurate and is to be
discouraged.