Compute the local tuncation error for the scheme and judge

Question

Question 1 - For the I.V.P of ODE y' = (t-1)e^-y, y(1) = 0, find an approximation to y(1.2) using the following numerical methods with Δt = 0.1. Compare the numerical solution with the exact solution and compute the error (use 5-digit rounding for all your calculations. Attach any computer code you need to solve the problem).

(a) Adams-Bashforth 2nd order method

(b) Trapezoidal Rule

(c) 2nd Order Backward Difference Formula

Question 2 - Suppose certain numerical method for ODE y' = f(t, y) has the form: Y_k+2 = Y_k+1 + Δt(5/12 f_k+2 + 2/3 f_k+1 - 1/12 f_k).

(a) Compute the Local Tuncation Error for the scheme and judge if the scheme is consistent.

(b) Based on the eigenvalue calculation, judge if the scheme is zero-stable.

(c) Apply the scheme to the standard test problem and plot the region of absolute stability for the scheme (please attach both the plot and the computer code to generate the plot).