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Consider the difference scheme λ Uk,j+1 = Uk,j + ;(−Uk−2,j + 16Uk−1‚j 12 - — 30Uk‚j + 16Uk+1‚j — Uk+2,j) for a heat equation ut = a²uxx, where λ = , and Ukj a² At u(xk, tj) Use the Fourier analysis method (let 4x² Uk,j = û³ eiwxk) to find the condition on 2, such that this scheme is stable. (A trig identity you might need: cos(20) = 2 cos² 0 - 1)

Consider the difference scheme λ Uk,j+1 = Uk,j + ;(−Uk−2,j + 16Uk−1‚j 12 - — 30Uk‚j + 16Uk+1‚j — Uk+2,j) for a heat equation ut = a²uxx, where λ = , and Ukj a² At u(xk, tj) Use the Fourier analysis method (let 4x² Uk,j = û³ eiwxk) to find the condition on 2, such that this scheme is stable. (A trig identity you might need: cos(20) = 2 cos² 0 - 1)

Advanced Engineering Mathematics
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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Consider the difference scheme λ Uk.j+1 = Uk,j + (−Uk-2.j + 16Uk-1,j − 30Uk,j + 16Uk+1,j — Uk+2,j) 12 a² At 4x² for a heat equation ut = a²uxx, where λ = and Uk‚j ≈ u(xk, tj) Use the Fourier analysis method (let Uk‚j û¹ eiwxk) to find the condition on 2, such that this scheme is stable. (A trig identity you might need: cos(20) = 2 cos² 0 − 1) =
Transcribed Image Text:Consider the difference scheme λ Uk.j+1 = Uk,j + (−Uk-2.j + 16Uk-1,j − 30Uk,j + 16Uk+1,j — Uk+2,j) 12 a² At 4x² for a heat equation ut = a²uxx, where λ = and Uk‚j ≈ u(xk, tj) Use the Fourier analysis method (let Uk‚j û¹ eiwxk) to find the condition on 2, such that this scheme is stable. (A trig identity you might need: cos(20) = 2 cos² 0 − 1) =
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