Neumann Stability Analysis Assume a difference scheme in the form (6.32) w(+1) j = l dl w() and make use of the Fourier transform

Neumann Stability Analysis Assume a difference scheme in the form (6.32) w(ν+1) j =  l dl w(ν)

and make use of the Fourier transform (6.17) w(ν) j = n −1 k=0 c (ν) k eikηjΔx for η = 2π nΔx .

a) What are the coefficients dl for the FTCS method (6.16)?

b) Prove linear independence n −1 k=0 αk exp[i 2π n kj]=0 =⇒ αk = 0 for all k Hint: FFT equivalence (C1.8).

c) Show c (ν+1) k = c (ν) k  l dl eikηlΔx .