16.920J/SMA 5212 Numerical Methods for PDEs 0 0 Slide 10 STABILITY ANALYSIS Coupled oDEs to Uncoupled ODEs Starting fre Au+b Premultiplication by E yield e-ld e-l4u+E-b AEE)b+E-b E=(EAEE-b+E-b STABILITY ANALYSIS Coupled oDEs to Uncoupled ODEs Continuing from di AE-u+E-6 Letu=E-i and F=E-B,we have16.920J/SMA 5212 Numerical Methods for PDEs 9 Slide 10 STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs Starting from du Au b dt = + ✟ ✟✠✟ 1 Premultiplication by E yields − 1 du 1 1 E E Au E b dt − − − = + ￾ ￾ ￾ Slide 11 STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs Continuing from 1 du 1 1 E E u E b dt − − − = Λ + ￾ ￾ ￾ 1 1 Let U E u and F E b, we have − − = = ✁ ✁ ✁ ✁ 1 2 1 where N λ λ λ − ✂ ✄ ☎ ✆ ☎ ✆ ☎ ✆ Λ = ☎ ✆ ☎ ✆ ☎ ✆ ✝ ✞ 0 0 ( ) 1 du 1 1 1 E E A EE u E b dt − − − − = + ✟ ✟ ✟ I Λ ( ) 1 du 1 1 1 E E AE E u E b dt − − − − = + ✠ ✠ ✠