Scientific Computing

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CS205 Homework #8 Problem 1 Give a criterion for the well-posedness of the k th order, scalar, homogeneous, constant-coefficient ODE u ( k ) + c k - 1 u ( k - 1) + · · · + c 1 u + c 0 u = 0 (Hint: Transform to a first-order system y = Ay and observe A is a matrix we’ve encountered previously in homework 3 problem 2) Problem 2 Consider the system of linear ODE’s y 1 y 2 t = 1 - 2 - 2 1 y 1 y 2 1. Consider the initial value problem with the above ode and the initial values y 1 (0) = y 2 (0) = 1 Show that the analytic solution to this initial value problem is y 1 ( t ) = y 2 ( t ) = e - t 2. If we use an integration method (such as Forward/Backward Euler, or trapezoidal rule) to compute the solution to this ODE numerically, will we get the same asymptotic behavior as the analytic solution as t → ∞ ? Problem 3 Consider the equation of motion for a simple, damped, 1D oscillator (a zero rest length spring in 1D with damping) F ( x, v
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Unformatted text preview: is the spring constant, b the (constant) damping coefficient, v = x t the velocity and a = v t = x tt the acceleration. 1. Show that this 2nd order ODE is equivalent to the 1st order linear system of ODEs ± x v ² t = ± 1-k m-b m ²± x v ² 2. Assume that we are using Forward Euler to solve this system numerically, with a timestep equal to Δ t . If λ 1 , λ 2 ∈ C are the complex eigenvalues of the matrix ± 1 Δ t-k Δ t m 1-b Δ t m ² show that the condition for stability is k λ 1 k < 1 and k λ 2 k < 1 3. Show that if b 2 < 4 km (such spring systems are referred to as under-damped ), then the eigenvalues of the matrix above are given as λ 1 , 2 = 1-b Δ t 2 m ± i Δ t 2 m p 4 km-b 2 4. Show that if b 2 < 4 km the condition for stability is Δ t < b/k . 2...
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