Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 68 / 3 / b / Solution

Apply the exponential-symbol order criterion for a multistep method, which follows by substituting the bilateral shift operator $e^{h{\partial }_t}$ into the exact-solution residual. Direct expansion givesAll coefficients through degree three vanish. If $\alpha \ne -5$, the fourth-degree coefficient is nonzero, so the normalized local truncation error has exact order three. At $\alpha =-5$, the fourth-degree coefficient vanishes and the fifth-degree coefficient is $1/10$, which is nonzero. ThusThis is the formal order of the full recurrence. The order-four member is not zero-stable, so it is not convergent. All the convergent members from part (a) have order three. At the degenerate value $\alpha =1$, the full recurrence still has formal order three, while cancellation of its common factor gives Backward Euler method of order one; that cancellation changes the method rather than its order calculation.