16.920J/SMA 5212 Numerical Methods for PDEs Predictor-Corrector Time Discretization Consider the numerical stability of the following predictor-corrector time discretization du l=-1+ as applied to the typical modal equation dt of the parabolic PDE. Substituting- and -into the predictor-corrector scheme i+l=u+h(aur+aeth)where t=nAt=nh "+ Utilizing the shift operator and rearranging the equations into matrix form, we obtai (1+Ah)s S n hS To determine the characteristic polynomial, set -(1+Mh P(a)=P(S)=1 (1+)SS16.920J/SMA 5212 Numerical Methods for PDEs 24 Predictor-Corrector Time Discretization Consider the numerical stability of the following predictor-corrector time discretization scheme 1 1 1 1 1 ˆ 1 ˆ ˆ 2 n n n n n n n n n du u u h dt du u u u h dt + + + + + = + ￾ ✁ = + + ✂ ✄ ☎ ✆ as applied to the typical modal equation t u ae dt du µ = λ + of the parabolic PDE. Substituting dt du and dt duˆ into the predictor-corrector scheme yields ( ) ( ) 1 1 1 1 ( 1) ˆ where 1 ˆ ˆ 2 n n n hn n n n n h n u u h u ae t n t nh u u u h u ae µ µ λ λ + + + + + = + + = ∆ = ✝ ✞ = + + + ✟ ✠ Utilizing the shift operator 1 1 ˆ ˆ n n n n Su u Su u + + = = and rearranging the equations into matrix form, we obtain ( ) ( ) 1 ˆ 1 1 1 1 2 2 2 n hn n S h h u ae h S S u hS µ λ λ ✡ ☛ − + ✡ ☛ ✡☞☛ ✌ ✍ ✌ ✍ = ✌ ✍ ✌ ✍ ✌ ✍ − + − ✎☞✏ ✌ ✍ ✌ ✍ ✎ ✏ ✎ ✏ To determine the characteristic polynomial, set ( ) ( ) ( ) ( ) 1 1 1 0 1 2 2 S h S h S S λ σ λ − + Ρ = Ρ = = − + −