Multistep methods reuse previous values and slopes. They are efficient, but their stability theory has an additional root condition.
Linear multistep method
Definition. A linear multistep method has the form \(\sum_{j=0}^k \alpha_j y_{n+j}=h\sum_{j=0}^k\beta_j f(t_{n+j},y_{n+j})\). It is explicit if \(\beta_k=0\) and implicit otherwise.
Consistency
Definition. Consistency requires the method to reproduce constant solutions and first derivatives correctly. In coefficient form this imposes \(\rho(1)=0\) and \(\rho'(1)=\sigma(1)\), where \(\rho,\sigma\) are the characteristic polynomials.
Zero-stability
Theorem. A linear multistep method is zero-stable exactly when every root of \(\rho\) lies in the closed unit disk and roots on the unit circle are simple. This prevents parasitic modes from growing when \(h\to0\).
Dahlquist equivalence
Theorem. For linear multistep methods, consistency plus zero-stability is equivalent to convergence. This is the ODE analogue of the consistency-stability-convergence pattern.
Adams and BDF families
Remark. Adams methods are built from polynomial interpolation of the right-hand side. Backward differentiation formulas approximate \(\mathrm{d}y/\mathrm{d}t\) using backward differences and are important for stiff equations.
Linear multistep form
A \(k\)-step method is \[ \sum_{j=0}^k\alpha_jy_{n+j} =h\sum_{j=0}^k\beta_jf_{n+j}. \] Define characteristic polynomials \[ \rho(\zeta)=\sum_{j=0}^k\alpha_j\zeta^j, \qquad \sigma(\zeta)=\sum_{j=0}^k\beta_j\zeta^j. \] Consistency requires \[ \rho(1)=0,\qquad \rho'(1)=\sigma(1). \]
Zero stability
Theorem (root condition). A linear multistep method is zero stable if and only if every root of \(\rho\) lies in \(|\zeta|\le1\), and every root on the unit circle is simple.
The homogeneous perturbation recurrence has modes \(n^\ell\zeta^n\). Roots outside the unit disc grow exponentially; repeated unit-modulus roots grow polynomially. The root condition is exactly uniform boundedness over a fixed time interval as \(h\to0\).
Theorem (Dahlquist equivalence). A consistent linear multistep method is convergent if and only if it is zero stable.
The proof writes the global error recurrence as the stable propagation of local truncation errors. Necessity follows by applying convergence to perturbed starting values and the zero differential equation.
Adams and BDF families
Adams--Bashforth methods integrate an interpolant through past derivative values and are explicit. Adams--Moulton includes the new derivative and is implicit. Backward differentiation formulas differentiate an interpolant through solution values: \[ \sum_{j=0}^k\alpha_jy_{n+j}=h f_{n+k}. \] BDF orders one and two are A-stable; orders three through six have large stability sectors; BDF is not zero stable above order six.
Starting and variable steps
A \(k\)-step method needs \(k-1\) starting values, usually from a Runge--Kutta method of comparable order. Low-order starts can pollute global accuracy. Variable step sizes change coefficients and invalidate fixed-step root analysis; practical solvers control step ratios and use Nordsieck vectors to represent scaled derivatives consistently.
Predictor--corrector and error control
An explicit Adams predictor can initialize an implicit Adams--Moulton corrector. Their difference estimates local error, but repeated correction is not equivalent to solving the implicit equation exactly unless converged. Modern variable-order solvers select both step and order based on estimated work per accuracy.