Use the Fourier analysis to show that the Richardson's method U¡‚j+1 − Uį‚j−1 _ a² (Ui+1‚j − 2U¡‚j + Ui−1,j) 24t (Δx)2 is unconditionally unstable. (Hint: This is a 2-step method, after using the Fourier analysis, you will get ûj+¹ = (coeff1) û¡ + (coeƒƒ2)û¡−¹, then use the analysis for multi-step methods (Sec 5.10) to write out its characteristic polynomial: r² = (coeff1)r + (coeff2). Then use this quadratic equation to determine if it satisfies the root condition.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
7. Use the Fourier analysis to show that the Richardson's method
Ui‚j+1 − U₁‚j−1 __a² (Ui+1,j − 2Uį,j + Ui-1,j)
2At
(Δx)2
is unconditionally unstable.
(Hint: This is a 2-step method, after using the Fourier analysis, you will get
ûj+¹ = (coeff1) û¡ + (coeƒƒ2) û¡−1, then use the analysis for multi-step methods (Sec 5.10) to write
out its characteristic polynomial: r² = (coeff1)r + (coeff2). Then use this quadratic equation to
determine if it satisfies the root condition.)
Transcribed Image Text:7. Use the Fourier analysis to show that the Richardson's method Ui‚j+1 − U₁‚j−1 __a² (Ui+1,j − 2Uį,j + Ui-1,j) 2At (Δx)2 is unconditionally unstable. (Hint: This is a 2-step method, after using the Fourier analysis, you will get ûj+¹ = (coeff1) û¡ + (coeƒƒ2) û¡−1, then use the analysis for multi-step methods (Sec 5.10) to write out its characteristic polynomial: r² = (coeff1)r + (coeff2). Then use this quadratic equation to determine if it satisfies the root condition.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,