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HW1 - MAE 290B Winter 2010 HOMEWORK 1 Due Mon in class...

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MAE 290B, Winter 2010 HOMEWORK 1 Due Mon 01-18-2010 in class Provide source codes used to solve all problems PROBLEM 1 Consider the generalized family of Euler schemes u n +1 = u n + αF ( u n , t n ) + βF ( u n +1 , t n +1 ) Δ t with α + β = 1. 1. Show that the condition α + β = 1 makes these schemes at least first-order accurate. 2. Show that the Implicit Euler method ( α = 0 , β = 1) is the most -stable member of this family while the Crank-Nicolson method ( α = β = 1 / 2) is the most accurate one. PROBLEM 2 Consider the ODE set d tt u - u + 3 d t u + 5 w = t, d t w - 20 u + 10 w = 0 , (1) with initial conditions u (0) = 0, d t u (0) = - 1, w (0) = 1. 1. Find the exact analytical solution of this problem. 2. Determine the maximum value of the time step Δ t max for which this ODE set can be integrated using the Explicit Euler numerical scheme. 3. Integrate numerically this ODE set for 0 < t < 10 using Δ t = 2Δ t max using the Explicit Euler, Implicit Euler and Crank Nicolson schemes.

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