16.920J/SMA 5212 Numerical Methods for PDEs Reminde Recall that we are considering a typical modal equation which had been obtained from the original equation EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator =An+aemn→tm+1-2h1n2--1=2ha(e加m 2h Solution of u consists of the complementary solution c", and the particular solution p",i.e There are several ways of solving for the complementary and particular solutions. One way is through use of the shift operator S and characteristic polynomial The time shift operator S operates on c" such that Sc"=c"+ S2c"=S(Se")=So Slide 25 EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator The complementary solution c" satisfies the homogenous equation c+-2h/c"-C-=0 hic 1616.920J/SMA 5212 Numerical Methods for PDEs 16 Reminder Recall that we are considering a typical modal equation which had been obtained from the original equation du Au b dt = + ￾ ￾✁￾ EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator ( ) 1 1 1 1 2 2 2 n n u u n hn n n n hn u ae u h u u ha e h µ µ λ λ + − − + − = + ✂ − − = Solution of u consists of the complementary solution n c , and the particular solution n p , i.e. n n n u = c + p There are several ways of solving for the complementary and particular solutions. One way is through use of the shift operator S and characteristic polynomial. The time shift operator S operates on n c such that n n 1 Sc c + = ( ) 2 n n n 1 n 2 S c S Sc Sc c + + = = = Slide 25 EXAMPLE 2 Leapfrog Time Discretization: Time Shift Operator The complementary solution n c satisfies the homogenous equation 1 1 2 0 2 0 n n n n n n c h c c c Sc h c S λ λ + − − − = − − =