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  1. Consider the IVP:

y0 = y            0 t 1           y(0) = 1.   (1)

(a) Use Euler Method to fifind an approximation to y(0.5) for this IVP with ∆t = 0.5.

(b) Reduce ∆t by half (∆t = 0.25) and compute the corresponding approximation to y(0.5).

(c) Use Richardson extrapolation and (a)-(b) to produce a more accurate approximation to y(0.5).

  1. Consider the initial value problem:

y 00 2y0 + y = tet 1 0 t 1,       (2)

y(0) = y0 (0) = 0.    (3)

(a) Transform this problem into an IVP for a fifirst order system of ODEs.

(b) Take the time step size ∆t = 0.1. Do one step of midpoint method


(c) Using (b), compute an approximation of the solution at t = 0.3 with the explicit 2-step Adams method.

  1. Consider the 2-stage RK method

(a) Prove that this method is convergent to the solution of the IVP y0 = f(t, y),0 < t T, y(0) = α (you may assume f is continuous and Lipschitz in y).

(b) Find the stability function R(z) for this RK method and prove that

R(z) = ez + O(z3).     (8)

Infer from (8) the order of the method.

(c) Find the interval of A-stability of this method. This is the intersection of its A-stability region S with the real (x) axis.

  1. Consider the 2-step method

yn+1 = yn + 1/3∆t(3f n 2f n1).    (9)

Discuss D-stability, consistency, and convergence for this method.

  1. Consider the method

yn+1 = yn + ∆t /10[5f n+1 + 4f n f n1].   (10)

(a) What type of method is it?

(b) Is it convergent?

  1. Consider the multi-step method

yn+1 = 5yn1 4yn + ∆t[4f n + 2f n1].   (11)

Discuss consistency, D-stability, and convergence.

  1. Find the region of A-stability (linear stability) of (a) the forward Euler method and (b) the backward Euler method.
  1. Write the RK methods corresponding to the following Butcher tableaux in Tables 1-2.

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