U UI lle orsandla (5.4.4). a Calculate the first nonzero term in the Taylor seri

Question

U UI lle orsandla (5.4.4). a Calculate the first nonzero term in the Taylor series of the truncation error Tf(h) for the finite difference formula defined by the second row of Table 5.2. Order of | Node location accuracy 0 h 2h 3h 4h -1 1 - 2 -; - 3 - |- 4 - 3 4 - Table 5.2. Weights for forward finite difference formulas (p = 0 in (5.4.2)). The values given here are for approximating the derivative at zero. See the text about the analogous backward differences where q = 0. The term order of accuracy is explained in Section 5.5.

Explanation / Answer

f(xi) ~ f(xi+1) -f(xi) /h

Truncation error = f(xi+1) -f(xi) /h - f '(xi)

expanding f(xi+1) in a taylor series around xi

f(xi+1) = f(xi) +  f '(xi)h + 1/2 f ''(xi)h2  + o(h3)

Tf(h) = 1/2 f ''(xi)h + o(h2)

= 1/2[ { f(xi+2) -2 f(xi+1) + f(xi) }/h2 ]h + o(h2)

= 1/2[{ f(2h) - 2 f(h) + f(0) }/h2 ]h + o(h2)

= -3/h + o(h2)

First non zero term is ( -3/h )

Related Questions

Navigate

• Browse (All)
• Browse U
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.