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.)
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
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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.)
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