16.920J/SMA 5212 Numerical Methods for PDEs When h increases from zero, decreases from 1. 0. As h continues to increase .o reaches a minimum of 0.5 with Ah=-l and then increases. As h increases further. o returns to 1.0 with /h=-2. Prior to this point, the scheme is stable. Increasing h and thus o beyond this point renders the scheme unstable Hence, this predictor-corrector scheme is stable for small h's and unstable for large h's; the limit for stability is /h=-2(from above) In general, we can analyze the absolute stability diagram for the predictor-corrector time discretization method in terms of G:(h)=1+2h+()2 h:Ah=-1±√2σ-1 A, the eigenvalue(s)of the A matrix can take on complex forms depending on the overning equation(as opposed to negative real values for the l-D parabolic PDE with central differencing for the spatial derivative) RELATIONSHIP BETWEEN o AND ah =o^h) Thus far, we have obtained the stability criterion of the time discretization scheme using a typical modal equation. We can generalize the relationship between o and /h as follows Starting from the set of coupled odes Au+b dt Apply a specific time discretization scheme like the leapfrog time discretizat Example 2 dt 2h16.920J/SMA 5212 Numerical Methods for PDEs 26 When h increases from zero, σ decreases from 1.0. As h continues to increase, σ reaches a minimum of 0.5 with λh = −1 and then increases. As h increases further, σ returns to 1.0 with λh = −2. Prior to this point, the scheme is stable. Increasing h and thus σ beyond this point renders the scheme unstable. Hence, this predictor-corrector scheme is stable for small h’s and unstable for large h’s; the limit for stability is λh = −2 (from above). In general, we can analyze the absolute stability diagram for the predictor-corrector time discretization method in terms of 2 ( ) : ( ) 1 2 h h h λ σ σ λ = + λ + or λh : λh = −1± 2σ −1 λ, the eigenvalue(s) of the A matrix can take on complex forms depending on the governing equation (as opposed to negative real values for the 1-D parabolic PDE with central differencing for the spatial derivative). RELATIONSHIP BETWEEN σ AND λh σ = σ(λh) Thus far, we have obtained the stability criterion of the time discretization scheme using a typical modal equation. We can generalize the relationship between σ and λh as follows: • Starting from the set of coupled ODEs du Au b dt = + ￾ ￾✁￾ • Apply a specific time discretization scheme like the leapfrog time discretization as in Example 2 1 1 2 n n du u u dt h + − − = Slide 39