16.920J/SMA 5212 Numerical Methods for PDEs EXAMPLE 1 Discrete Time Operator =EAE-·EAE-·ENE IL eae-Iald were a =4 0 l1"=451c1+252c2 where are constants l2=A"521c+"k2c2 Alternative view Alternatively one can view the solution as Ur U=A"U where U=E-i EXAMPLE 1 Comparison Comparing the solution of the semi-discretized problem where time is kept continuous l2 to the solution where time is discretized16.920J/SMA 5212 Numerical Methods for PDEs 14 EXAMPLE 1 Discrete Time Operator As 1 0 1 2 0 where = 0 n n n n n u E E u λ λ − ￾ ✁ = Λ Λ ✂☎ ✆✄ ✟ ✝✞✝✟ ' ' 1 1 11 1 2 12 2 ' ' 2 1 21 1 2 22 2 n n n n n n u c c u c c λ ξ λ ξ λ ξ λ ξ = + = + 1 1 0 2 ' where are constants. ' c E u c − ✠☛✡ = ☞ ✌ ✍☛✎ ✏✞✏✑ Slide 21 Alternative View Alternatively, one can view the solution as: 0 1 1 1 2 0 2 2 n n n n U U U U λ λ ✒ ✓ ✒ ✓ ✒ ✓ = ✔ ✕ ✔ ✕ ✖ ✗ ✖ ✗ ✖ ✗ 0 1 where n n U U U E u − = Λ = ✘ ✘ ✘ ✘ EXAMPLE 1 Comparison Comparing the solution of the semi-discretized problem where time is kept continuous [ ] 1 2 1 11 12 1 2 2 21 22 t t u e c c u e λ λ ξ ξ ξ ξ ✙✛✚ ✙ ✚ ✙ ✚ = ✜ ✢ ✜✣✛✤✢ ✜✣ ✤✢ ✣ ✤ to the solution where time is discretized [ ] 1 11 12 1 1 2 2 21 22 2 ' ' n n n u c c u ξ ξ λ ξ ξ λ ✥ ✦ ✥✛✦ ✥ ✦ = ✧ ★ ✧✩✛✪★ ✧✩ ✪★ ✧ ★ ✩ ✪ 1 1 1 1 0 , .... n A E E u E E E E E E u − − − − = Λ ✭ = Λ ⋅ Λ ⋅ ⋅ Λ ⋅ ✫✬✫✭ A A A