16.920J/SMA 5212 Numerical Methods for PDEs The difference equation where time is continuous has exponential solution e The difference equation where time is discretized has power Slide 22 EXAMPLE 1 Comparison In equivalence, the transient solution of the difference equation must decay with time. i.e for this particular form of time discretization EXAMPLE 2 Leapfrog Time Discretization Consider a typical modal equation of the form where a, is the eigenvalue of the associated matrix A ( For simplicity, we shall henceforth drop the subscript j) We shall apply the"leapfrog time discretization scheme given as where h=△ 2h Substituting into the modal equation yield16.920J/SMA 5212 Numerical Methods for PDEs 15 difference equation where time is continuous has exponential solution The . t e λ The difference equation where time is discretized has power solution . n λ Slide 22 EXAMPLE 1 Comparison In equivalence, the transient solution of the difference equation must decay with time, i.e. 1 n λ < for this particular form of time discretization. Slide 23 EXAMPLE 2 Leapfrog Time Discretization Consider a typical modal equation of the form t j du u ae dt µ λ ￾ ✁ = + ✂ ✄ ☎ ✆ where is the eigenvalue of the associated matrix . λ j A (For simplicity, we shall henceforth drop the subscript j). We shall apply the “leapfrog” time discretization scheme given as 1 1 where 2 n n du u u h t dt h + − − = = ∆ Substituting into the modal equation yields 1 1 2 n n u u h + − − ( ) t t nh u ae µ λ = = + n hn u ae µ = λ + Slide 24