16.920J/SMA 5212 Numerical Methods for PDEs EXAMPLE 1 Continuous Time Operator Proceeding as before, or otherwise(solving the ODEs directly) we can obtain the solution where M and n are eigenvalues of A and eigenvectors pertaining to n, and n, respectively As the transient solution must decay with time, it is imperative that Rcal(4)≤0for Slide g EXAMPLE 1 Discrete Time Operator Suppose we have somehow discretized the time operator on the LHS to obtain a1l1+a12l2 l2=a21l1+a2l2 where the subscript n stands for the n time level, then where u=a”l2 nd a Since A is independent of time u Au = AAu =.=a"u In later examples, we shall apply specific time discretization schemes such as the"leapfrog " and Euler fonvard time discretization schemes16.920J/SMA 5212 Numerical Methods for PDEs 13 EXAMPLE 1 Continuous Time Operator Proceeding as before, or otherwise (solving the ODEs directly), we can obtain the solution 1 2 1 2 1 1 11 2 12 2 1 21 2 22 t t t t u c e c e u c e c e λ λ λ λ ξ ξ ξ ξ = + = + 11 21 1 2 21 22 1 2 where and are eigenvalues of and and are eigenvectors pertaining to and respectively. A ξ ξ λ λ ξ ξ λ λ ￾✂✁ ￾ ✁ ✄ ☎ ✄ ☎ ✆✂✝ ✆ ✝ ( j) As the transient solution must decay with time, it is imperative that Real λ ≤ 0 for j =1, 2. Slide 19 EXAMPLE 1 Discrete Time Operator Suppose we have somehow discretized the time operator on the LHS to obtain 1 1 1 11 1 12 2 1 1 2 21 1 22 2 n n n n n n u a u a u u a u a u − − − − = + = + where the subscript n stands for the n th time level, then 1 11 12 1 2 21 22 where and n n n T n n a a u Au u u u A a a − ✞ ✟ = = ✞ ✟ = ✠ ✡ ☛ ☞ ☛ ☞ ✌ ✌ ✌ Since A is independent of time, 1 2 0 .... n n n n u Au AAu A u − − = = = = ✍ ✍ ✍ ✍ In later examples, we shall apply specific time discretization schemes such as the “leapfrog” and Euler-forward time discretization schemes. Slide 20