16.920J/SMA 5212 Numerical Methods for PDEs EXAMPLE 2 Leapfrog Time Discretization In particular, by applying to the 1-D Parabolic PDE au d4 at the central difference scheme for spatial discretization, we obtain -2 which is the tridiagonal matrix Slide 30 EXAMPLE 2 eapfrog Time Discretization According to analysis of a general tridiagonal matrix B(a, b, c), the lvalues of b are b+2 N△x2 The mostdangerous "mode is that associated with the eigenvalue of largest magnitude a1(b)=h+√2mn2+1 O2 which can be plotted in the absolute stability diagram16.920J/SMA 5212 Numerical Methods for PDEs 19 EXAMPLE 2 Leapfrog Time Discretization In particular, by applying to the 1-D Parabolic PDE 2 2 u u t x υ ∂ ∂ = ∂ ∂ the central difference scheme for spatial discretization, we obtain which is the tridiagonal matrix Slide 30 EXAMPLE 2 Leapfrog Time Discretization According to analysis of a general triadiagonal matrix B(a,b,c), the eigenvalues of B are 2 2 cos , 1,..., 1 2 2cos j j j b ac j N N j N x π λ π υ λ ￾ ✁ = + = − ✂ ✄ ☎ ✆ ✝ ￾ ✁ ✞ = − + ✂ ✄ ✟ ✠ ∆ ✡ ☎ ✆ ☛ The most “dangerous” mode is that associated with the eigenvalue of largest magnitude max 2 4 x υ λ = − ∆ i.e. ( ) ( ) 2 2 max 1 max max 2 2 max 2 max max 1 1 h h h h h h σ λ λ λ σ λ λ λ = + + = − + which can be plotted in the absolute stability diagram. 2 2 1 1 2 1 1 1 2 A x υ −☞ ✌ ✍ ✎ − ✍ ✎ ✍ ✎ = ✍ ✎ ∆ ✍ ✎ ✍ ✎ ✍ ✎ − ✍ ✎ ✏ ✑ 0 0