16.920J/SMA 5212 Numerical Methods for PDEs STABILITY ANALYSIS Discretization We obtain at x dt Ar(uo-2u+"2) dax2(4-2n2+n) 动=a2(1-21+1 N-1 (a2-2ux=1+lx) dt Note that we need not evaluate u at x=x, and x=XN since u, and u, are given as boundary conditions Side 6 STABILITY ANALYSIS Matrix Formulation Assembling the system of equations, we obtain dt 0 dhu,△x2 dt 0 1-2 DUN Slide 7 716.920J/SMA 5212 Numerical Methods for PDEs 7 STABILITY ANALYSIS Discretization We obtain at 1 1 2 1 2 : ( 2 ) o du x u u u dt x υ = − + ∆ 2 2 2 1 2 3 : ( 2 ) du x u u u dt x υ = − + ∆ 2 1 1 : ( 2 ) j j j j j du x u u u dt x υ = − − + + ∆ 1 1 2 2 1 : ( 2 ) N N N N N du x u u u dt x − υ − = − − − + ∆ 0 0 Note that we need not evaluate at and since and are given as boundary conditions. N N u x x x x u u = = Slide 6 STABILITY ANALYSIS Matrix Formulation Assembling the system of equations, we obtain Slide 7 1 1 2 2 2 2 1 1 2 2 1 0 1 2 1 0 1 2 1 1 0 1 2 o j j N N N du u u dt x du u dt du x u dt u du u x dt υ υ υ − − ￾ ✁ ￾ ✁ − ✂ ✄ ￾ ✁ ￾ ✁ ✂ ✄ ∆ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ − ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ = + ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ∆ − ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✂ ✄ ✂ ✄ ✂ ✄ ✂ ✂ ✄ ✂ ✄ − ✂ ✄ ☎ ✆✝☎ ✆ ✂ ✂ ∆☎ ✆ ✂☎ ✆✄ ✄ ✄ ✄ 0 ✄ 0 A