Question

determine the constants a, b, c in the differentiation formula y '' (x0) =ay(x0-h) +by(x0) +cy(x0 +h). so that the method of the highest possible order and the error term of the method.

Answer 1

Answer to Question #74135, Math / Quantitative Methods

Determine the constants $a, b, c$ in the differentiation formula

$$y ^ {\prime \prime} (x) = a y (x - h) + b y (x) + c y (x + h)$$

so that the method of the highest possible order and the error term of the method.

Solution.

We choose the coefficients $a, b, c$ so that $y''(x)$ is exact for $y(x) = x^j$ for $j = 0,1,2,\ldots$

From Taylor series:

$$y (x - h) = y (x) - h y ^ {\prime} (x) + \frac {1}{2 !} h ^ {2} y ^ {\prime \prime} (x) - \frac {1}{3 !} h ^ {3} y ^ {\prime \prime \prime} (x) + \frac {1}{4 !} h ^ {4} y ^ {(4)} (x) + \dots$$ $$y (x + h) = y (x) + h y ^ {\prime} (x) + \frac {1}{2 !} h ^ {2} y ^ {\prime \prime} (x) + \frac {1}{3 !} h ^ {3} y ^ {\prime \prime \prime} (x) + \frac {1}{4 !} h ^ {4} y ^ {(4)} (x) + \dots$$

If $y(x) = 1$ then

$$a + b + c = 0$$

If $y(x) = x$ then

$$h (a - c) = 0$$

If $y(x) = x^2$ then

$$\frac {h ^ {2} (a + c)}{2} = 1$$

Thus:

$$a = c = \frac {1}{h ^ {2}}; b = - \frac {2}{h ^ {2}}$$

The error:

$$\varepsilon = y ^ {\prime \prime} (x) - D _ {h} ^ {(2)} y (x)$$ $$D _ {h} ^ {(2)} y (x) = \frac {y (x - h) - 2 y (x) + y (x + h)}{h ^ {2}}$$

Using the Taylor series approach:

Answer to Question #74135, Math / Quantitative Methods

$$\varepsilon \approx y ^ {\prime \prime} (x) - \frac {y (x - h) - 2 y (x) + y (x + h)}{h ^ {2}} = y ^ {\prime \prime} (x) - y ^ {\prime \prime} (x) - 2 \cdot \frac {1}{4 !} h ^ {2} y ^ {(4)} (x)$$ $$\varepsilon \approx - \frac {2}{2 4} h ^ {2} y ^ {(4)} (x) = - \frac {h ^ {2}}{1 2} y ^ {(4)} (x)$$

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