Understanding Finite Difference Schemes for Derivatives

To compute the derivative of a continuous function, especially when an analytical solution is difficult or impossible, we often use numerical methods. Finite difference schemes are a common approach, approximating derivatives using function values at discrete points. The accuracy of these approximations depends on the order of the scheme. In this problem, we need to find the formula for the third derivative, $f'''(x_i)$, using a fourth-order central finite divided difference scheme with a step length 'h'.

Taylor Series Expansion and Central Differences

The foundation of finite difference methods lies in Taylor series expansions. A Taylor series expands a function around a point $x_i$:

\(f(x_i + nh) = f(x_i) + nhf'(x_i) + \frac{{(nh)^2}}{{2!}}f''(x_i) + \frac{{(nh)^3}}{{3!}}f'''(x_i) + \frac{{(nh)^4}}{{4!}}f^{(4)}(x_i) + \frac{{(nh)^5}}{{5!}}f^{(5)}(x_i) + \frac{{(nh)^6}}{{6!}}f^{(6)}(x_i) + \frac{{(nh)^7}}{{7!}}f^{(7)}(x_i) + \mathcal{O}(h^8)\)

For central difference approximations of odd derivatives (like the first or third derivative), we typically combine Taylor series expansions for points symmetrically located around \(x_i\), such as \(f(x_i + nh)\) and \(f(x_i - nh)\).

Let's consider the expansion for \(f(x_i + nh) - f(x_i - nh)\):

\(f(x_i + nh) - f(x_i - nh) = 2 \left[ nhf'(x_i) + \frac{{(nh)^3}}{{3!}}f'''(x_i) + \frac{{(nh)^5}}{{5!}}f^{(5)}(x_i) + \frac{{(nh)^7}}{{7!}}f^{(7)}(x_i) + \mathcal{O}(h^9) \right]\)

A fourth-order central difference formula for the third derivative will involve symmetric points around \(x_i\) up to \(x_{i \pm 3h}\) (i.e., \(x_{i-3}, x_{i-2}, x_{i-1}, x_{i+1}, x_{i+2}, x_{i+3}\)). The general form for the third derivative is often a linear combination of these symmetric differences:

\(f'''(x_i) \approx \frac{1}{{h^3}} \left[ A_3 (f(x_{i+3h}) - f(x_{i-3h})) + A_2 (f(x_{i+2h}) - f(x_{i-2h})) + A_1 (f(x_{i+h}) - f(x_{i-h})) \right]\)

Our goal is to find the coefficients \(A_1, A_2, A_3\) such that when these terms are expanded using Taylor series, the \(f'''(x_i)\) term is isolated and lower order derivative terms (\(f'(x_i)\), \(f''(x_i)\), etc.) and higher odd derivative terms (\(f^{(5)}(x_i)\)) cancel out to achieve fourth-order accuracy.

Deriving the Fourth-Order Central Difference Formula for Third Derivative

Let's examine the structure of the correct formula provided, which is:

\(f'''\left( {{x_i}} \right) = \frac{{ - f\left( {{x_{i + 3}}} \right) + 8f\left( {{x_{i + 2}}} \right) - 13f\left( {{x_{i + 1}}} \right) + 13f\left( {{x_{i - 1}}} \right) - 8f\left( {{x_{i - 2}}} \right) + f\left( {{x_{i - 3}}} \right)}}{{8{h^3}}}\)

We can rearrange the numerator by grouping symmetric terms:

Numerator \(= - (f(x_{i+3h}) - f(x_{i-3h})) + 8(f(x_{i+2h}) - f(x_{i-2h})) - 13(f(x_{i+h}) - f(x_{i-h}))\)

Now, let's substitute the Taylor series expansion for \(f(x_{i+nh}) - f(x_{i-nh})\) into this grouped numerator:

1. For \(-(f(x_{i+3h}) - f(x_{i-3h}))\):
   \(= -2 \left[ 3h f'(x_i) + \frac{{(3h)^3}}{{3!}} f'''(x_i) + \frac{{(3h)^5}}{{5!}} f^{(5)}(x_i) + \frac{{(3h)^7}}{{7!}} f^{(7)}(x_i) + \dots \right]\)
   \(= -2 \left[ 3h f'(x_i) + \frac{{27h^3}}{6} f'''(x_i) + \frac{{243h^5}}{120} f^{(5)}(x_i) + \frac{{2187h^7}}{5040} f^{(7)}(x_i) + \dots \right]\)
2. For \(8(f(x_{i+2h}) - f(x_{i-2h}))\):
   \(= 8 \times 2 \left[ 2h f'(x_i) + \frac{{(2h)^3}}{{3!}} f'''(x_i) + \frac{{(2h)^5}}{{5!}} f^{(5)}(x_i) + \frac{{(2h)^7}}{{7!}} f^{(7)}(x_i) + \dots \right]\)
   \(= 16 \left[ 2h f'(x_i) + \frac{{8h^3}}{6} f'''(x_i) + \frac{{32h^5}}{120} f^{(5)}(x_i) + \frac{{128h^7}}{5040} f^{(7)}(x_i) + \dots \right]\)
3. For \(-13(f(x_{i+h}) - f(x_{i-h}))\):
   \(= -13 \times 2 \left[ h f'(x_i) + \frac{{h^3}}{{3!}} f'''(x_i) + \frac{{h^5}}{{5!}} f^{(5)}(x_i) + \frac{{h^7}}{{7!}} f^{(7)}(x_i) + \dots \right]\)
   \(= -26 \left[ h f'(x_i) + \frac{{h^3}}{6} f'''(x_i) + \frac{{h^5}}{120} f^{(5)}(x_i) + \frac{{h^7}}{5040} f^{(7)}(x_i) + \dots \right]\)

Now, let's sum the coefficients for each derivative term:

• Coefficient for \(f'(x_i)\):
  \((-2 \times 3h) + (16 \times 2h) + (-26 \times h) = -6h + 32h - 26h = 0h\)
  This confirms that the first derivative term cancels, which is expected for central difference approximations of odd derivatives.
• Coefficient for \(f'''(x_i)\):
  \(-2 \left( \frac{{27h^3}}{6} \right) + 16 \left( \frac{{8h^3}}{6} \right) - 26 \left( \frac{{h^3}}{6} \right)\)
  \(= \frac{{h^3}}{6} (-54 + 128 - 26) = \frac{{h^3}}{6} (48) = 8h^3\)
  This term correctly gives \(8h^3 f'''(x_i)\).
• Coefficient for \(f^{(5)}(x_i)\):
  \(-2 \left( \frac{{243h^5}}{120} \right) + 16 \left( \frac{{32h^5}}{120} \right) - 26 \left( \frac{{h^5}}{120} \right)\)
  \(= \frac{{h^5}}{120} (-486 + 512 - 26) = \frac{{h^5}}{120} (0) = 0h^5\)
  The fifth derivative term cancels, which is crucial for achieving a fourth-order accurate scheme.
• Coefficient for \(f^{(7)}(x_i)\):
  \(-2 \left( \frac{{2187h^7}}{5040} \right) + 16 \left( \frac{{128h^7}}{5040} \right) - 26 \left( \frac{{h^7}}{5040} \right)\)
  \(= \frac{{h^7}}{5040} (-4374 + 2048 - 26) = \frac{{h^7}}{5040} (-2352)\)
  This is the first non-zero higher-order term, indicating the error is related to \(h^7\).

So, the numerator evaluates to:

\(\text{Numerator} = 8h^3 f'''(x_i) - \frac{{2352}}{{5040}} h^7 f^{(7)}(x_i) + \mathcal{O}(h^9)\)

When divided by \(8h^3\):

\(f'''(x_i) = \frac{{8h^3 f'''(x_i) - \frac{{2352}}{{5040}} h^7 f^{(7)}(x_i) + \mathcal{O}(h^9)}}{{8h^3}}\)

\(f'''(x_i) = f'''(x_i) - \frac{{2352}}{{40320}} h^4 f^{(7)}(x_i) + \mathcal{O}(h^6)\)

The error term is \(\mathcal{O}(h^4)\), confirming that this is a fourth-order accurate scheme.

Conclusion

The correct formula for computing the third derivative \(f'''(x_i)\) using the fourth-order central finite divided difference scheme with step length \(h\) is:

\(f'''\left( {{x_i}} \right) = \frac{{ - f\left( {{x_{i + 3}}} \right) + 8f\left( {{x_{i + 2}}} \right) - 13f\left( {{x_{i + 1}}} \right) + 13f\left( {{x_{i - 1}}} \right) - 8f\left( {{x_{i - 2}}} \right) + f\left( {{x_{i - 3}}} \right)}}{{8{h^3}}}\)

This matches the first option provided.