16.920J/SMA 5212 Numerical Methods for PDEs Hence for eachi, j=1, 2,.N-l h Slide fl RELATIONSHIP BETWEEN o AND ah =o(h) Note that the analysis performed above is identical to the analysis carried out using the modal equation du =AU+F All the analysis carried out earlier for a single modal equation is applicable to the matrix after the appropriate manipulation to obtain an uncoupled set of odes Each equation can be solved independently for Ur and the U,"'s can then be coupled through E =EU Slide 4 RELATIONSHIP BETWEEN o AND ah =o(h) Hence, applying any"consistent numerical technique to each equation in the set of coupled linear ODEs is mathematically equivalent to Integrating each equation in the uncoupled set, 3. Re-coupling the results to form the final solution These 3 steps are commonly referred to as the ISOLATION THEOREM Slide 4316.920J/SMA 5212 Numerical Methods for PDEs 28 Hence for each j, j = 1,2,….,N−1, 1 2 j j j S S U F h λ − ￾ ✁ − − = − ✂ ✄ ☎ ✆ Slide 41 RELATIONSHIP BETWEEN σ AND λh σ = σ(λh) Note that the analysis performed above is identical to the analysis carried out using the modal equation j dU U F dt λ ✝ ✞ = + ✟ ✠ ✡ ☛ All the analysis carried out earlier for a single modal equation is applicable to the matrix after the appropriate manipulation to obtain an uncoupled set of ODEs. Each equation can be solved independently for and the 's can then be coupled through . th n n n n j j j U U u = EU☞ ☞ Slide 42 RELATIONSHIP BETWEEN σ AND λh σ = σ(λh) Hence, applying any “consistent” numerical technique to each equation in the set of coupled linear ODEs is mathematically equivalent to 1. Uncoupling the set, 2. Integrating each equation in the uncoupled set, 3. Re-coupling the results to form the final solution. These 3 steps are commonly referred to as the ISOLATION THEOREM Slide 43