 Linear multistep method

A linear multistep method approximates an ordinary differential equation $y^{\prime }=f(t,y)$ using values retained from several time levels. On a uniform grid $t_n=t_0+nh$, its constant-coefficient form isThe linearity is in the way values and derivatives are combined; the vector field $f$ itself can be nonlinear. The method is explicit when ${\beta }_k=0$ and implicit when the new value appears inside $f$ through ${\beta }_k\ne 0$. An implicit update requires a suitable algebraic solve, in addition to a formal recurrence.^[ref]

The two-step Adams-Bashforth method, for example, uses $Y_{n+1}=Y_n+h(3f_n-f_{n-1})/2$. It reuses the preceding derivative rather than calculating several new Runge-Kutta stages within one step. A $k$-step recurrence needs $k$ starting values, often supplied by a Runge-Kutta method of compatible accuracy.

Consistency controls the defect on a smooth exact solution; zero-stability controls the amplification of starting and local errors. Both are required by the Dahlquist equivalence theorem for convergence under its regularity assumptions. Changing the step sizes generally requires changing the coefficients and revisiting stability; uniform-grid coefficients cannot simply be reused at arbitrary time spacings.

Linear multistep methods are numerical techniques used to solve ordinary differential equations (ODEs) by approximating the solutions at discrete points. Unlike single-step methods (like the Euler method or Runge-Kutta methods) that only use information from the current time step to compute the next step, linear multistep methods utilize information from multiple previous time steps.