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Unformatted text preview: Numerical Methods for Initial Value Problems Consider the IVP ( ! ) du dt = f ( u ) , u ( t = 0) = u In one-step methods, we will approximate du dt ≈ u n +1- u n Δ t ≈ ¯ f, u n +1 ≈ u n + Δ t ¯ f where ¯ f is an approximation to the RHS of the IVP ( ! ). Forward Euler The forward Euler method is u n +1 = u n + Δ tf ( u n ) Forward Euler is an explicit method, and is first-order accurate and condi- tionally stable. For the linear case du/dt = u = au , the exact solution is u ( t ) = u e at . The growth factor for forward Euler is defined by u n +1 = (1 + a Δ t ) u n ≡ Gu n The exact growth factor H = e a Δ t . Note that G = e a Δ t to first order ( con- sistency ). For n timesteps, u n = G n u = (1 + a Δ t ) n u → e na Δ t u = e at n u = u ( t n ) as n → ∞ and Δ t → 0 with n Δ t = t n held fixed: we have proved that u n converges to the exact solution u ( t n ) in this limit. (Henceforth we will prove a numerical method is consistent and stable; then convergence follows from the Equivalence Theorem.) To analyze stability , we consider the model problem du dt = au, a < 0 ( model problem ) because then the exact solution u ( t ) = e-| a | Δ t u decays. u n = G n u must not grow for a < 0, so stability requires | G | ≤ 1. For forward Euler we have- 1 ≤ 1 + a Δ t = 1-| a | Δ t ≤ 1 or Δ t ≤ 2 / | a | for stability. To find the order of accuracy for the full nonlinear problem ( ! ), we cal- culate the LTE = Δ...
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- Fall '11