16.920J/SMA 5212 Numerical Methods for PDEs STABILITY ANALYSIS PDE to Coupled ODEs Or in compact form du lu+b where u=[a1u2…… △x2 We have reduced the I-D PDe to a set of Coupled OdEs! Slide 8 STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A If A is a nonsingular matrix, as in this case, it is then possible to find a set of eigenvalues 元={4,2,…,…,不- A-11=0 For each eigenvalue 1. we can evaluate the eigenvector p consisting of a set of mesh point values v/,i.e y=Y vy STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A The(N-1)x(N-1)matrix E formed by the(N-1)columns r diagonalizes the matrix A by E-AE=A16.920J/SMA 5212 Numerical Methods for PDEs 8 STABILITY ANALYSIS PDE to Coupled ODEs Or in compact form We have reduced the 1-D PDE to a set of Coupled ODEs! Slide 8 STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A If A is a nonsingular matrix, as in this case, it is then possible to find a set of eigenvalues λ = {λ1 ,λ2 ,....,λ j ,....,λN −1} from det( A− λI ) = 0. For each eigenvalue , we can evaluate the eigenvector consisting of a set of mesh point values , i.e. j j j i V v λ Slide 9 STABILITY ANALYSIS Eigenvalue and Eigenvector of Matrix A The ( 1) ( 1) matrix formed by the ( 1) columns diagonalizes the matrix by j N N E N V A − × − − 1 E AE − = Λ where [ 1 2 1 ] T N u u u u = − ￾ 2 2 0 0 0 T o N u u b x x υ υ ✁ ✂ = ✄ ☎ ∆ ∆ ✆ ✝ ✞ du Au b dt = + ✟ ✟✠✟ 1 2 1 Tj j j j V N v v v − ✡ ☛ = ☞ ✌