4.In attempting to solve a simple PDE, a system of finite-difference equations of the formun+1j=1+νν001+νν-ν01+νunj.Investigate the stability of the scheme.For the stability, we need||A|| ≤1 or that|eig(A)| ≤1. To compute the eigenvalues weneed to solve1 +ν−λν001 +ν−λν−ν01 +ν−λ= 0or(1 +ν−λ)3−ν3= 0So1 +ν−λ=ννe2πi/3νe4πi/3λ=11 +ν1−e2πi/31 +ν1−e4πi/3Note that the last two eigenvalues are complex conjugate of each other.Now clearly theabsolute value of the first is 1. The absolute value of the other two is the same and it is
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