16.920J/SMA 5212 Numerical Methods for PDEs EXAMPLE 3 Euler-Forward time discretization stability diagram The stability diagram for the Euler-forward time discretization in the /h-plane is Unit Circle Ah) Region of Stability EXAMPLE 3 Euler-Forward Time Discretization: Absolute Stability Diagram As applied to the 1-D Parabolic PDE, n=Am leaves the unit circle at /h=-2 dath=△t=0 o with h increasing Re(o) The stability limit for largest h=4t=-2 gleaves the unit circle at g=-1, i.e. o=l+/h=-I nh=-2 h ince it is the extreme lide 3816.920J/SMA 5212 Numerical Methods for PDEs 23 EXAMPLE 3 Euler-Forward Time Discretization: Stability Diagram The stability diagram for the Euler-forward time discretization in the λh-plane is Slide 37 EXAMPLE 3 Euler-Forward Time Discretization: Absolute Stability Diagram As applied to the 1-D Parabolic PDE, max 2 4 x υ λ λ − = = ∆ max 2 The stability limit for largest h t λ − ≡ ∆ = σ leaves the unit circle at σ = −1, i.e. σ = 1+ λh = −1 max 2 λh 2 h λ − = − ￾ = since it is the extreme. Slide 38 Im(λh) -2 0 Re(λh) Region of Stability -1 Unit Circle Im(σ ) -1 1 Re(σ ) σ at h = ∆t = 0 σ leaves the unit circle at λh = −2 σ with h increasing