16.920J/SMA 5212 Numerical Methods for PDEs the criterion for stability of the space discretization of a paraone ga One may note that a is always real and negative, thereby satisfyin PDE, keeping time continuous ide 3/ EXAMPLE 2 Leapfrog Time Discretization: Absolute Stability Diagram for o As applied to the 1-D Parabolic PDE, the absolute stability diagram for o is Im(o) Region of nstability Unit circle g with h o with h G1ath=△r=0 Increasing Increasing Reo G2ath=△t=0 Stability In this case, o and o, start out being on the unit circle(h=At=0). However, the spurious root (refer to following slide) leaves the unit circle as h starts increasing Therefore, the spurious root causes the leapfrog time discretization scheme to be unstable, irrespective of how small h=At is, although it does not affect the accuracy The leapfrog time discretization for the I-D Diffusion Equation is unstable Slide 3216.920J/SMA 5212 Numerical Methods for PDEs 20 One may note that λ j is always real and negative, thereby satisfying the criterion for stability of the space discretization of a parabolic PDE, keeping time continuous. Slide 31 EXAMPLE 2 Leapfrog Time Discretization: Absolute Stability Diagram for σ As applied to the 1-D Parabolic PDE, the absolute stability diagram for σ is In this case, σ1 and σ 2 start out being on the unit circle (h = ∆t = 0). However, the spurious root (refer to following slide) leaves the unit circle as h starts increasing. Therefore, the spurious root causes the leapfrog time discretization scheme to be unstable, irrespective of how small h = ∆t is, although it does not affect the accuracy. The leapfrog time discretization for the 1-D Diffusion Equation is unstable. Slide 32 Im(σ ) Re(σ ) Unit circle σ1 with h increasing σ2 with h increasing Region of Instability Region of σ2 at h = ∆t = 0 Stability σ1 at h = ∆t = 0