Answer the following questions: (a) To what order accuracy in Ax does this finite-difference scheme approximate f'(x)? 1 -(ƒ(x + 4^x) − f (x − 4^x)) 8Ax (b) Approximately how much more accurate is the following scheme than the scheme in (a)? 1 4Ax −(ƒ(x+2^x) − f (x − 2^x)) (c) Suppose you wish to increase the accuracy of your finite-difference differentiation scheme. Can you obtain arbitrarily small error by decreasing the step size Ax? Why or why not?
Answer the following questions: (a) To what order accuracy in Ax does this finite-difference scheme approximate f'(x)? 1 -(ƒ(x + 4^x) − f (x − 4^x)) 8Ax (b) Approximately how much more accurate is the following scheme than the scheme in (a)? 1 4Ax −(ƒ(x+2^x) − f (x − 2^x)) (c) Suppose you wish to increase the accuracy of your finite-difference differentiation scheme. Can you obtain arbitrarily small error by decreasing the step size Ax? Why or why not?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Finite-Difference Schemes: An Examination of Accuracy**
**Answer the following questions:**
(a) To what order accuracy in Δx does this finite-difference scheme approximate \( f'(x) \)?
\[
\frac{1}{8\Delta x} \left( f(x + 4\Delta x) - f(x - 4\Delta x) \right)
\]
(b) Approximately how much more accurate is the following scheme than the scheme in (a)?
\[
\frac{1}{4\Delta x} \left( f(x + 2\Delta x) - f(x - 2\Delta x) \right)
\]
(c) Suppose you wish to increase the accuracy of your finite-difference differentiation scheme. Can you obtain arbitrarily small error by decreasing the step size Δx? Why or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6e2f997-9120-4975-9388-a1bc7e4c3a16%2Fb1cd6e83-e2a1-447b-8a14-7a2d831b1c3a%2Ff499y4_processed.png&w=3840&q=75)
Transcribed Image Text:**Finite-Difference Schemes: An Examination of Accuracy**
**Answer the following questions:**
(a) To what order accuracy in Δx does this finite-difference scheme approximate \( f'(x) \)?
\[
\frac{1}{8\Delta x} \left( f(x + 4\Delta x) - f(x - 4\Delta x) \right)
\]
(b) Approximately how much more accurate is the following scheme than the scheme in (a)?
\[
\frac{1}{4\Delta x} \left( f(x + 2\Delta x) - f(x - 2\Delta x) \right)
\]
(c) Suppose you wish to increase the accuracy of your finite-difference differentiation scheme. Can you obtain arbitrarily small error by decreasing the step size Δx? Why or why not?
Expert Solution
Step 1
By Taylor series expansion, we have
We have to find the order of accuracy in of the finite-difference approximate scheme for
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
