16.920J/SMA 5212 Numerical Methods for PDEs P(a)=P(S)=SS-1-h-13h21=0 0( trivial root) =1+nh+-A2h2 the scheme is a one-root method. Compared to +Ah+-A2h2+ the scheme is second-order accurate To obtain the particular solution, one can perform a matrix inversion and obtain 1+h with the complementary solution being B"=月1+m+h2 4 The absolute stability diagram(showing 1=--)for the 1-D Parabolic PDE is Region of Instability h increasing further Stability16.920J/SMA 5212 Numerical Methods for PDEs 25 ( ) ( ) 2 2 2 2 1 1 0 2 0 (trivial root) 1 1 2 S S S h h h h σ λ λ σ σ λ λ ￾ ✁ Ρ = Ρ = − − − = ✂ ✄ ☎ ✆ ✝ = = + + i.e. the scheme is a one-root method. Compared to 1 2 2 1 .... 2 h e h h λ = + λ + λ + the scheme is second-order accurate. To obtain the particular solution, one can perform a matrix inversion and obtain ( ) 2 2 2 1 1 1 2 1 e h h ahe e h p h hn h n λ λ λ µ µ µ − − − + + = with the complementary solution being n n n c h h ✟✞ ✠ ✡☛☞ = = + + 2 2 2 1 βσ β 1 λ λ The absolute stability diagram (showing 2 4 ∆x = − υ λ ) for the 1-D Parabolic PDE is Im(σ ) -1 1 Re(σ ) 0.5 Region of Stability Region of Instability h increasing from 0 h increasing further