4. (The Saul’yev ADE Method)

Implement this method for the heat equation as a variation of the discussion of the Barakat and Clark method in Section 13.4. In this case we use a single array to capture the approximate solution at each time level. The array is computed at uneven steps (n = 1, 3, 5,…) using equation (13.29) and at even steps (n = 0, 2, 4,…) using equation (13.30). Use a single array for both the upward and downward sweeps. It will always hold the approximate solution at all time levels up to and including expiration.

Compare the accuracy and run-time efficiency of this method with those of the Barakat and Clark ADE method and the Crank–Nicolson scheme.