Compute the local tuncation error for the scheme and judge

Question

Question 1. Use Taylor Expansion to compute the local truncation error for the implicit Trapezoidal method: Y^k+1 = Y^k + ?t/2 (f^k + f^k+1)

Question 2. Suppose certain numerical method for ODE y ′ = f(t, y) has the form:

Y^k+2 = Y^k+1 + ?t/12 (-f^k + 8f^k+1 + 5f^k+2).

(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.