16.920J/SMA 5212 Numerical Methods for PDEs RELATIONSHIP BETWEEN o AND ah =o(h) The above set of odes becomes +b 2h Introducing the time shift operator S Su==+haui +2hb A |i"=-b Premultiplying e on the lhs and rhs and introducing I=EE operating on ui" EAE-E EE e-b RELATIONSHIP BETWEEN o AND ah S-S we obtain A-E E h 2h which is a set of uncoupled equations16.920J/SMA 5212 Numerical Methods for PDEs 27 RELATIONSHIP BETWEEN σ AND λh σ = σ(λh) • The above set of ODEs becomes 1 1 2 n n n u u n Au b h + − − = + ￾ ￾ ￾ ￾ • Introducing the time shift operator S 1 2 2 2 n n n n n n u Su hAu hb S S S A I u b h − = + + ✁ ✂ − − = − ✄ ☎ ✆ ✝ ✞ ✞ ✞ ✞ ✞ ✞ • 1 1 1 1 1 2 S S n E AE E E E u E b h − − − − − ✟ ✠ − − = − ✡ ☛ ☞ ✌ ✍ ✍ Slide 40 RELATIONSHIP BETWEEN σ AND λh σ = σ(λh) • Putting 1 1 , n n n n U E u F E b − − ✎ = ✎ = ✎ ✎ we obtain 1 1 2 S S n n E E U F h − − ✏ ✑ − Λ − = − ✒ ✓ ✔ ✕ ✖ ✖ i.e. 1 2 S S n n U F h − ✗ ✘ − Λ − = − ✙ ✚ ✛ ✜ ✢ ✢ which is a set of uncoupled equations. 1 1 Premultiplying on the LHS and RHS and introducing operating on n E I EE u − − = ✣ Λ 1 2 S S h − −