Suppose we have an equispaced grid z, xo + jh (0 ≤ i ≤n) and given a function f, denote fj:= f(x₂). (a) For 1 ≤j≤n-1, write the expression of the quadratic interpolant II(z) of a function f through the points (-1,-1), (j, f) and (a,+1, fj+1). (b) Evaluate II'(-1), II'(,) and II'(+1) to recover the second-order accurate FD schemes for the first derivative (centered, right-sided, left-sided). (c) Evaluate II"(z) to recover the first-order accurate FD scheme for the second derivative.
Suppose we have an equispaced grid z, xo + jh (0 ≤ i ≤n) and given a function f, denote fj:= f(x₂). (a) For 1 ≤j≤n-1, write the expression of the quadratic interpolant II(z) of a function f through the points (-1,-1), (j, f) and (a,+1, fj+1). (b) Evaluate II'(-1), II'(,) and II'(+1) to recover the second-order accurate FD schemes for the first derivative (centered, right-sided, left-sided). (c) Evaluate II"(z) to recover the first-order accurate FD scheme for the second derivative.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. Suppose we have an equispaced grid a
xo + jh (0 ≤ i ≤ n) and given a function f, denote
(a) For 1 ≤j≤n-1, write the expression of the quadratic interpolant II (a) of a function f through
the points (-1, fj-1), (j, fj) and (x+1, fj+1).
(b) Evaluate II'(x-1), II'(,) and II'(x+1) to recover the second-order accurate FD schemes for
the first derivative (centered, right-sided, left-sided).
(e) Evaluate II" (z) to recover the first-order accurate FD scheme for the second derivative.
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