16.920J/SMA 5212 Numerical Methods for PDEs EXAMPLE 3 Euler-Forward Time Discretization: Stability Analysis Analyze the stability of the explicit Euler-forward time discretization as applied to the modal equation du dt Substituting u=u"+h- where h= At into the modal equation, we obtain u"+-(1+/h)u=0 EXAMPLE 3 Euler-Forward Time Discretization: Stability Analysis Making use of the shift operator S n+-(1+Ah)c"=Sc-(1+h)l"=[S-(1+h)k"=0 characteristic polynom Therefo o(h)=1+h The euler-forward time discretization scheme is stable if ≡1+Ah<1 or bounded by Ah=0-1 S.t.ok<I in the ah lide 3616.920J/SMA 5212 Numerical Methods for PDEs 22 EXAMPLE 3 Euler-Forward Time Discretization: Stability Analysis Analyze the stability of the explicit Euler-forward time discretization n 1 n du u u dt t + − = ∆ as applied to the modal equation du u dt = λ 1 1 Substituting where into the modal equation, we obtain (1 ) 0 n n n n du u u h h t dt u λh u + + = + = ∆ − + = Slide 35 EXAMPLE 3 Euler-Forward Time Discretization: Stability Analysis Making use of the shift operator S 1 (1 ) (1 ) [ (1 )] 0 n n n n n c λh c Sc λh c S λh c + − + = − + = − + = Therefore ( ) 1 and n n h h c σ λ λ βσ = + = The Euler-forward time discretization scheme is stable if σ ≡1+ λh <1 or bounded by λh =σ −1 s.t. σ <1 in the λh-plane. Slide 36 characteristic polynomial