Multistep Methods

Multistep methods use information from several previous steps to advance the solution. They achieve high order with fewer function evaluations per step than Runge-Kutta methods but require careful initialization and have restricted stability regions.

Adams Methods

Definition7.4Adams-Bashforth Methods

The $k$ -step Adams-Bashforth (AB) method is an explicit multistep method: $y_{n+1} = y_n + h \sum_{j=0}^{k-1} b_j f(t_{n-j}, y_{n-j})$ where the coefficients $b_j$ are determined by requiring exactness for polynomials of degree $\leq k - 1$ . Key instances:

AB1 (Euler): $y_{n+1} = y_n + hf_n$
AB2: $y_{n+1} = y_n + \frac{h}{2}(3f_n - f_{n-1})$
AB3: $y_{n+1} = y_n + \frac{h}{12}(23f_n - 16f_{n-1} + 5f_{n-2})$
AB4: $y_{n+1} = y_n + \frac{h}{24}(55f_n - 59f_{n-1} + 37f_{n-2} - 9f_{n-3})$

Definition7.5Adams-Moulton Methods

The $k$ -step Adams-Moulton (AM) method is implicit: $y_{n+1} = y_n + h \sum_{j=-1}^{k-1} b_j f(t_{n-j}, y_{n-j})$ (note $f_{n+1}$ appears). Key instances:

AM0 (backward Euler): $y_{n+1} = y_n + hf_{n+1}$
AM1 (trapezoidal): $y_{n+1} = y_n + \frac{h}{2}(f_{n+1} + f_n)$
AM2: $y_{n+1} = y_n + \frac{h}{12}(5f_{n+1} + 8f_n - f_{n-1})$

AM methods are one order higher than AB methods with the same number of steps, and have larger stability regions.

Predictor-Corrector Methods

ExampleAdams-Bashforth-Moulton Pair

A predictor-corrector pair uses AB $k$ to predict $y_{n+1}^P$ (explicit), evaluates $f(t_{n+1}, y_{n+1}^P)$ , then applies AM $(k-1)$ to correct: $y_{n+1}^C = y_n + h\sum b_j f_{n-j}$ using $f_{n+1} \approx f(t_{n+1}, y_{n+1}^P)$ . The AB4/AM3 pair (PECE mode): predict with AB4, evaluate $f_{n+1}^P$ , correct with AM3, evaluate $f_{n+1}^C$ . This gives fourth-order accuracy with only 2 function evaluations per step.

BDF Methods

Definition7.6Backward Differentiation Formulas

The $k$ -step BDF method is: $\sum_{j=0}^k \alpha_j y_{n+1-j} = h \beta_0 f(t_{n+1}, y_{n+1})$ (only $f_{n+1}$ appears on the right). BDF methods are designed for stiff problems:

BDF1 (backward Euler): $y_{n+1} - y_n = hf_{n+1}$
BDF2: $\frac{3}{2}y_{n+1} - 2y_n + \frac{1}{2}y_{n-1} = hf_{n+1}$
BDF $k$ for $k \leq 6$ are stable; BDF7 and higher are unstable

BDF methods are A( $\alpha$ )-stable: stable in a sector $|\arg(-\lambda h)| < \alpha$ where $\alpha$ decreases from $90°$ (BDF1-2) to $\sim 18°$ (BDF6).

RemarkStiff ODE Solvers in Practice

Modern stiff ODE solvers (LSODE, VODE, CVODE, MATLAB's ode15s) are variable-order, variable-step BDF or NDF (numerical differentiation formulas) implementations. They automatically adjust the order $k$ (between 1 and 5) and step size $h$ based on error and stability estimates. Newton iteration solves the implicit equations, with the Jacobian $\partial f/\partial y$ updated only when convergence slows, amortizing the $O(n^3)$ factorization cost.