Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.

Definitions

Numerical methods for ordinary differential equations approximate solutions to initial value problems of the form

$${\displaystyle y'=f(t,y),\quad y(t_{0})=y_{0}.}$$

The result is approximations for the value of ${\displaystyle y(t)}$ at discrete times ${\displaystyle t_{i))$:

$${\displaystyle y_{i}\approx y(t_{i})\quad {\text{where))\quad t_{i}=t_{0}+ih,}$$

${\displaystyle h}$

${\displaystyle \Delta t}$

${\displaystyle i}$

Multistep methods use information from the previous ${\displaystyle s}$ steps to calculate the next value. In particular, a linear multistep method uses a linear combination of ${\displaystyle y_{i))$ and ${\displaystyle f(t_{i},y_{i})}$ to calculate the value of ${\displaystyle y}$ for the desired current step. Thus, a linear multistep method is a method of the form

$${\displaystyle {\begin{aligned}&y_{n+s}+a_{s-1}\cdot y_{n+s-1}+a_{s-2}\cdot y_{n+s-2}+\cdots +a_{0}\cdot y_{n}\\&\qquad {}=h\cdot \left(b_{s}\cdot f(t_{n+s},y_{n+s})+b_{s-1}\cdot f(t_{n+s-1},y_{n+s-1})+\cdots +b_{0}\cdot f(t_{n},y_{n})\right)\\&\Leftrightarrow \sum _{j=0}^{s}a_{j}y_{n+j}=h\sum _{j=0}^{s}b_{j}f(t_{n+j},y_{n+j}),\end{aligned))}$$

${\displaystyle a_{s}=1}$

${\displaystyle a_{0},\dotsc ,a_{s-1))$

${\displaystyle b_{0},\dotsc ,b_{s))$

One can distinguish between explicit and implicit methods. If ${\displaystyle b_{s}=0}$, then the method is called "explicit", since the formula can directly compute ${\displaystyle y_{n+s))$. If ${\displaystyle b_{s}\neq 0}$ then the method is called "implicit", since the value of ${\displaystyle y_{n+s))$ depends on the value of ${\displaystyle f(t_{n+s},y_{n+s})}$, and the equation must be solved for ${\displaystyle y_{n+s))$. Iterative methods such as Newton's method are often used to solve the implicit formula.

Sometimes an explicit multistep method is used to "predict" the value of ${\displaystyle y_{n+s))$. That value is then used in an implicit formula to "correct" the value. The result is a predictor–corrector method.

Examples

Consider for an example the problem

$${\displaystyle y'=f(t,y)=y,\quad y(0)=1.}$$

${\displaystyle y(t)=e^{t))$

One-step Euler

A simple numerical method is Euler's method:

$${\displaystyle y_{n+1}=y_{n}+hf(t_{n},y_{n}).}$$

This method, applied with step size ${\displaystyle h={\tfrac {1}{2))}$ on the problem ${\displaystyle y'=y}$, gives the following results:

$${\displaystyle {\begin{aligned}y_{1}&=y_{0}+hf(t_{0},y_{0})=1+{\tfrac {1}{2))\cdot 1=1.5,\\y_{2}&=y_{1}+hf(t_{1},y_{1})=1.5+{\tfrac {1}{2))\cdot 1.5=2.25,\\y_{3}&=y_{2}+hf(t_{2},y_{2})=2.25+{\tfrac {1}{2))\cdot 2.25=3.375,\\y_{4}&=y_{3}+hf(t_{3},y_{3})=3.375+{\tfrac {1}{2))\cdot 3.375=5.0625.\end{aligned))}$$

Two-step Adams–Bashforth

Euler's method is a one-step method. A simple multistep method is the two-step Adams–Bashforth method

$${\displaystyle y_{n+2}=y_{n+1}+{\tfrac {3}{2))hf(t_{n+1},y_{n+1})-{\tfrac {1}{2))hf(t_{n},y_{n}).}$$

${\displaystyle y_{n+1))$

${\displaystyle y_{n))$

${\displaystyle y_{n+2))$

${\displaystyle y_{0}=1}$

${\displaystyle y_{1))$

$${\displaystyle {\begin{aligned}y_{2}&=y_{1}+{\tfrac {3}{2))hf(t_{1},y_{1})-{\tfrac {1}{2))hf(t_{0},y_{0})=1.5+{\tfrac {3}{2))\cdot {\tfrac {1}{2))\cdot 1.5-{\tfrac {1}{2))\cdot {\tfrac {1}{2))\cdot 1=2.375,\\y_{3}&=y_{2}+{\tfrac {3}{2))hf(t_{2},y_{2})-{\tfrac {1}{2))hf(t_{1},y_{1})=2.375+{\tfrac {3}{2))\cdot {\tfrac {1}{2))\cdot 2.375-{\tfrac {1}{2))\cdot {\tfrac {1}{2))\cdot 1.5=3.7812,\\y_{4}&=y_{3}+{\tfrac {3}{2))hf(t_{3},y_{3})-{\tfrac {1}{2))hf(t_{2},y_{2})=3.7812+{\tfrac {3}{2))\cdot {\tfrac {1}{2))\cdot 3.7812-{\tfrac {1}{2))\cdot {\tfrac {1}{2))\cdot 2.375=6.0234.\end{aligned))}$$

${\displaystyle t=t_{4}=2}$

${\displaystyle e^{2}=7.3891\ldots }$

Families of multistep methods

Three families of linear multistep methods are commonly used: Adams–Bashforth methods, Adams–Moulton methods, and the backward differentiation formulas (BDFs).

Adams–Bashforth methods

The Adams–Bashforth methods are explicit methods. The coefficients are ${\displaystyle a_{s-1}=-1}$ and ${\displaystyle a_{s-2}=\cdots =a_{0}=0}$, while the ${\displaystyle b_{j))$ are chosen such that the methods have order s (this determines the methods uniquely).

The Adams–Bashforth methods with s = 1, 2, 3, 4, 5 are (Hairer, Nørsett & Wanner 1993, §III.1; Butcher 2003, p. 103):

$${\displaystyle {\begin{aligned}y_{n+1}&=y_{n}+hf(t_{n},y_{n}),\qquad {\text{(This is the Euler method)))\\y_{n+2}&=y_{n+1}+h\left({\frac {3}{2))f(t_{n+1},y_{n+1})-{\frac {1}{2))f(t_{n},y_{n})\right),\\y_{n+3}&=y_{n+2}+h\left({\frac {23}{12))f(t_{n+2},y_{n+2})-{\frac {16}{12))f(t_{n+1},y_{n+1})+{\frac {5}{12))f(t_{n},y_{n})\right),\\y_{n+4}&=y_{n+3}+h\left({\frac {55}{24))f(t_{n+3},y_{n+3})-{\frac {59}{24))f(t_{n+2},y_{n+2})+{\frac {37}{24))f(t_{n+1},y_{n+1})-{\frac {9}{24))f(t_{n},y_{n})\right),\\y_{n+5}&=y_{n+4}+h\left({\frac {1901}{720))f(t_{n+4},y_{n+4})-{\frac {2774}{720))f(t_{n+3},y_{n+3})+{\frac {2616}{720))f(t_{n+2},y_{n+2})-{\frac {1274}{720))f(t_{n+1},y_{n+1})+{\frac {251}{720))f(t_{n},y_{n})\right).\end{aligned))}$$

The coefficients ${\displaystyle b_{j))$ can be determined as follows. Use polynomial interpolation to find the polynomial p of degree ${\displaystyle s-1}$ such that

$${\displaystyle p(t_{n+i})=f(t_{n+i},y_{n+i}),\qquad {\text{for ))i=0,\ldots ,s-1.}$$

$${\displaystyle p(t)=\sum _{j=0}^{s-1}{\frac {(-1)^{s-j-1}f(t_{n+j},y_{n+j})}{j!(s-j-1)!h^{s-1))}\prod _{i=0 \atop i\neq j}^{s-1}(t-t_{n+i}).}$$

${\displaystyle y'=f(t,y)}$

${\displaystyle y'=p(t)}$

$${\displaystyle y_{n+s}=y_{n+s-1}+\int _{t_{n+s-1))^{t_{n+s))p(t)\,\mathrm {d} t.}$$

${\displaystyle b_{j))$

$${\displaystyle b_{s-j-1}={\frac {(-1)^{j)){j!(s-j-1)!))\int _{0}^{1}\prod _{i=0 \atop i\neq j}^{s-1}(u+i)\,\mathrm {d} u,\qquad {\text{for ))j=0,\ldots ,s-1.}$$

${\displaystyle f(t,y)}$

The Adams–Bashforth methods were designed by John Couch Adams to solve a differential equation modelling capillary action due to Francis Bashforth. Bashforth (1883) published his theory and Adams' numerical method (Goldstine 1977).

Adams–Moulton methods

The Adams–Moulton methods are similar to the Adams–Bashforth methods in that they also have ${\displaystyle a_{s-1}=-1}$ and ${\displaystyle a_{s-2}=\cdots =a_{0}=0}$. Again the b coefficients are chosen to obtain the highest order possible. However, the Adams–Moulton methods are implicit methods. By removing the restriction that ${\displaystyle b_{s}=0}$, an s-step Adams–Moulton method can reach order ${\displaystyle s+1}$, while an s-step Adams–Bashforth methods has only order s.

The Adams–Moulton methods with s = 0, 1, 2, 3, 4 are (Hairer, Nørsett & Wanner 1993, §III.1; Quarteroni, Sacco & Saleri 2000) listed, where the first two methods are the backward Euler method and the trapezoidal rule respectively:

$${\displaystyle {\begin{aligned}y_{n}&=y_{n-1}+hf(t_{n},y_{n}),\\y_{n+1}&=y_{n}+{\frac {1}{2))h\left(f(t_{n+1},y_{n+1})+f(t_{n},y_{n})\right),\\y_{n+2}&=y_{n+1}+h\left({\frac {5}{12))f(t_{n+2},y_{n+2})+{\frac {8}{12))f(t_{n+1},y_{n+1})-{\frac {1}{12))f(t_{n},y_{n})\right),\\y_{n+3}&=y_{n+2}+h\left({\frac {9}{24))f(t_{n+3},y_{n+3})+{\frac {19}{24))f(t_{n+2},y_{n+2})-{\frac {5}{24))f(t_{n+1},y_{n+1})+{\frac {1}{24))f(t_{n},y_{n})\right),\\y_{n+4}&=y_{n+3}+h\left({\frac {251}{720))f(t_{n+4},y_{n+4})+{\frac {646}{720))f(t_{n+3},y_{n+3})-{\frac {264}{720))f(t_{n+2},y_{n+2})+{\frac {106}{720))f(t_{n+1},y_{n+1})-{\frac {19}{720))f(t_{n},y_{n})\right).\end{aligned))}$$

The derivation of the Adams–Moulton methods is similar to that of the Adams–Bashforth method; however, the interpolating polynomial uses not only the points ${\displaystyle t_{n-1},\dots ,t_{n-s))$, as above, but also ${\displaystyle t_{n))$. The coefficients are given by

$${\displaystyle b_{s-j}={\frac {(-1)^{j)){j!(s-j)!))\int _{0}^{1}\prod _{i=0 \atop i\neq j}^{s}(u+i-1)\,\mathrm {d} u,\qquad {\text{for ))j=0,\ldots ,s.}$$

The Adams–Moulton methods are solely due to John Couch Adams, like the Adams–Bashforth methods. The name of Forest Ray Moulton became associated with these methods because he realized that they could be used in tandem with the Adams–Bashforth methods as a predictor-corrector pair (Moulton 1926); Milne (1926) had the same idea. Adams used Newton's method to solve the implicit equation (Hairer, Nørsett & Wanner 1993, §III.1).

Backward differentiation formulas (BDF)

The BDF methods are implicit methods with ${\displaystyle b_{s-1}=\cdots =b_{0}=0}$ and the other coefficients chosen such that the method attains order s (the maximum possible). These methods are especially used for the solution of stiff differential equations.

Analysis

The central concepts in the analysis of linear multistep methods, and indeed any numerical method for differential equations, are convergence, order, and stability.

Consistency and order

The first question is whether the method is consistent: is the difference equation

$${\displaystyle {\begin{aligned}&a_{s}y_{n+s}+a_{s-1}y_{n+s-1}+a_{s-2}y_{n+s-2}+\cdots +a_{0}y_{n}\\&\qquad {}=h{\bigl (}b_{s}f(t_{n+s},y_{n+s})+b_{s-1}f(t_{n+s-1},y_{n+s-1})+\cdots +b_{0}f(t_{n},y_{n}){\bigr )},\end{aligned))}$$

${\displaystyle y'=f(t,y)}$

${\displaystyle y_{n+s))$

${\displaystyle y_{n+s-1},\ldots ,y_{n))$

${\displaystyle t_{n+s))$

$${\displaystyle \sum _{k=0}^{s-1}a_{k}=-1\quad {\text{and))\quad \sum _{k=0}^{s}b_{k}=s+\sum _{k=0}^{s-1}ka_{k}.}$$

If the method is consistent, then the next question is how well the difference equation defining the numerical method approximates the differential equation. A multistep method is said to have order p if the local error is of order ${\displaystyle O(h^{p+1})}$ as h goes to zero. This is equivalent to the following condition on the coefficients of the methods:

$${\displaystyle \sum _{k=0}^{s-1}a_{k}=-1\quad {\text{and))\quad q\sum _{k=0}^{s}k^{q-1}b_{k}=s^{q}+\sum _{k=0}^{s-1}k^{q}a_{k}{\text{ for ))q=1,\ldots ,p.}$$

${\displaystyle s+1}$

These conditions are often formulated using the characteristic polynomials

$${\displaystyle \rho (z)=z^{s}+\sum _{k=0}^{s-1}a_{k}z^{k}\quad {\text{and))\quad \sigma (z)=\sum _{k=0}^{s}b_{k}z^{k}.}$$

$${\displaystyle \rho (e^{h})-h\sigma (e^{h})=O(h^{p+1})\quad {\text{as ))h\to 0.}$$

${\displaystyle \rho (1)=0}$

${\displaystyle \rho '(1)=\sigma (1)}$

Stability and convergence

The numerical solution of a one-step method depends on the initial condition ${\displaystyle y_{0))$, but the numerical solution of an s-step method depend on all the s starting values, ${\displaystyle y_{0},y_{1},\ldots ,y_{s-1))$. It is thus of interest whether the numerical solution is stable with respect to perturbations in the starting values. A linear multistep method is zero-stable for a certain differential equation on a given time interval, if a perturbation in the starting values of size ε causes the numerical solution over that time interval to change by no more than Kε for some value of K which does not depend on the step size h. This is called "zero-stability" because it is enough to check the condition for the differential equation ${\displaystyle y'=0}$ (Süli & Mayers 2003, p. 332).

If the roots of the characteristic polynomial ρ all have modulus less than or equal to 1 and the roots of modulus 1 are of multiplicity 1, we say that the root condition is satisfied. A linear multistep method is zero-stable if and only if the root condition is satisfied (Süli & Mayers 2003, p. 335).

Now suppose that a consistent linear multistep method is applied to a sufficiently smooth differential equation and that the starting values ${\displaystyle y_{1},\ldots ,y_{s-1))$ all converge to the initial value ${\displaystyle y_{0))$ as ${\displaystyle h\to 0}$. Then, the numerical solution converges to the exact solution as ${\displaystyle h\to 0}$ if and only if the method is zero-stable. This result is known as the Dahlquist equivalence theorem, named after Germund Dahlquist; this theorem is similar in spirit to the Lax equivalence theorem for finite difference methods. Furthermore, if the method has order p, then the global error (the difference between the numerical solution and the exact solution at a fixed time) is ${\displaystyle O(h^{p})}$ (Süli & Mayers 2003, p. 340).

Furthermore, if the method is convergent, the method is said to be strongly stable if ${\displaystyle z=1}$ is the only root of modulus 1. If it is convergent and all roots of modulus 1 are not repeated, but there is more than one such root, it is said to be relatively stable. Note that 1 must be a root for the method to be convergent; thus convergent methods are always one of these two.

To assess the performance of linear multistep methods on stiff equations, consider the linear test equation y' = λy. A multistep method applied to this differential equation with step size h yields a linear recurrence relation with characteristic polynomial

$${\displaystyle \pi (z;h\lambda )=(1-h\lambda \beta _{s})z^{s}+\sum _{k=0}^{s-1}(\alpha _{k}-h\lambda \beta _{k})z^{k}=\rho (z)-h\lambda \sigma (z).}$$

Example

Consider the Adams–Bashforth three-step method

$${\displaystyle y_{n+3}=y_{n+2}+h\left({23 \over 12}f(t_{n+2},y_{n+2})-{4 \over 3}f(t_{n+1},y_{n+1})+{5 \over 12}f(t_{n},y_{n})\right).}$$

$${\displaystyle \rho (z)=z^{3}-z^{2}=z^{2}(z-1)}$$

${\displaystyle z=0,1}$

${\displaystyle z=1}$

The other characteristic polynomial is

$${\displaystyle \sigma (z)={\frac {23}{12))z^{2}-{\frac {4}{3))z+{\frac {5}{12))}$$

First and second Dahlquist barriers

These two results were proved by Germund Dahlquist and represent an important bound for the order of convergence and for the A-stability of a linear multistep method. The first Dahlquist barrier was proved in Dahlquist (1956) and the second in Dahlquist (1963).

First Dahlquist barrier

The first Dahlquist barrier states that a zero-stable and linear q-step multistep method cannot attain an order of convergence greater than q + 1 if q is odd and greater than q + 2 if q is even. If the method is also explicit, then it cannot attain an order greater than q (Hairer, Nørsett & Wanner 1993, Thm III.3.5).

Second Dahlquist barrier

The second Dahlquist barrier states that no explicit linear multistep methods are A-stable. Further, the maximal order of an (implicit) A-stable linear multistep method is 2. Among the A-stable linear multistep methods of order 2, the trapezoidal rule has the smallest error constant (Dahlquist 1963, Thm 2.1 and 2.2).