Questions: Multistep Methods: Adams-Bashforth and Adams-Moulton

Question 1

Why do Adams multistep methods require a 'startup procedure' using a single-step method like RK4?

• A. Adams-Bashforth is implicit and needs initial guesses at every step before it can begin
• B. A k-step Adams method needs k prior solution values, but at t₀ only y₀ is available — a single-step method generates y₁, ..., yₖ₋₁ first
• C. Single-step methods are more stable and must validate the initial data before multistep methods can proceed
• D. The Adams formula is only numerically valid after the transient behavior of the ODE has died out

Answer: (B)

The Adams-Bashforth k-step formula requires the function values f(tₙ,yₙ), f(tₙ₋₁,yₙ₋₁), ..., back k steps. At t₀ you only have y₀, so there is nothing in the history. A single-step method (typically RK4) must be run for the first k−1 steps to populate the history before the Adams formula can be applied. This startup cost is a fixed overhead that pays off on long integrations.

Question 2

A student argues that since Adams-Moulton is more accurate than Adams-Bashforth at the same order, you should simply use Adams-Moulton alone. What is the strongest counterargument?

• A. Adams-Moulton is less numerically stable than Adams-Bashforth for most ODEs
• B. Adams-Moulton is implicit — it requires solving for yₙ₊₁ at each step; using Adams-Bashforth to predict yₙ₊₁ first (predictor-corrector) achieves nearly the same accuracy with only explicit function evaluations
• C. Adams-Bashforth is always more accurate for smooth solutions
• D. Adams-Moulton cannot be applied to initial value problems, only to boundary value problems

Answer: (B)

Adams-Moulton includes f(tₙ₊₁, yₙ₊₁) in its formula, making it implicit — yₙ₊₁ appears on both sides. Solving this at each step requires a root-finding iteration (expensive). The predictor-corrector strategy uses Adams-Bashforth to predict yₙ₊₁, evaluates f there, and plugs that value into the Adams-Moulton corrector formula — giving implicit-level accuracy while requiring only one or two explicit function evaluations per step.

Question 3

A multistep Adams method is more efficient than RK4 for long smooth integrations because it reuses previously computed derivative values rather than computing new intermediate stages.

• T. True
• F. False

Answer: True

Classical RK4 requires 4 new function evaluations per step, every step, without reusing any prior information. Adams methods in predictor-corrector mode typically need 1-2 evaluations per step by reusing function values already stored from prior steps. For long integrations where the per-step savings multiply over thousands of steps, this efficiency advantage is substantial.

Question 4

Adams multistep methods are well-suited for problems that require frequent, large changes in step size during integration.

• T. True
• F. False

Answer: False

Changing step size in a multistep method is problematic: all stored past function values were computed at the old step size and cannot simply be reused at a new step size without reformulation. Step-size adaptation typically requires restarting the multistep formula, negating the efficiency gained. Single-step methods like RK4 handle variable step sizes naturally. Adams methods shine on long, smooth integrations at a fixed or slowly varying step size.

Question 5

What is the fundamental tradeoff that motivates Adams multistep methods over single-step Runge-Kutta methods, and when does that tradeoff favor multistep methods?

Model answer: Single-step methods compute new intermediate stages at every step and discard all accumulated history, requiring 4+ function evaluations per step. Multistep methods amortize this cost by storing past values of f(t, y) and reusing them in the next-step formula, reducing per-step evaluations to 1-2. The cost: startup (a single-step method must generate the first k values) and difficulty with variable step sizes. The tradeoff favors multistep methods for long, smooth integrations at a fixed step size — where the per-step savings compound — and disfavors them for stiff or highly variable problems requiring frequent adaptation.

This efficiency gain is the entire reason to use multistep methods. Once the startup phase is complete and the solution is smooth, a predictor-corrector Adams pair achieves the same accuracy per step as RK4 at roughly half the function evaluation cost. On an integration spanning 100,000 steps, that difference matters enormously.