16.920J/SMA 5212 Numerical Methods for PDEs di=au+F which is a set of Uncoupled ODEs STABILITY ANALYSIS Coupled oDEs to Uncoupled ODEs Expanding yields dU=U1+F due=au, +F2 d=4+F Since the equations are independent of one another, they be solved The idea then is to solve for u and determine i= eU STABILITY ANALYSIS Coupled oDEs to Uncoupled ODEs Considering the case of b independent of time, for the general j"equation is the solution forj=1, 2, . N-l16.920J/SMA 5212 Numerical Methods for PDEs 10 d U U F dt = Λ + ✁ ￾✁ ￾✁ which is a set of Uncoupled ODEs! Slide 12 STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs Expanding yields 1 1 1 1 dU U F dt = λ + 2 2 2 2 dU U F dt = λ + j j j j dU U F dt = λ + 1 1 1 1 N N N N dU U F dt λ − = − − + − Since the equations are independent of one another, they can be solved separately. The idea then is to solve for U and determine u = EU ✂ ✂ ✂ Slide 13 STABILITY ANALYSIS Coupled ODEs to Uncoupled ODEs Considering the case of independent of time, for the general equation, th b j ✄ jt 1 j j j j U c e F λ λ = − is the solution for j = 1,2,….,N−1