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

is an integer.

steps to calculate the next value. In particular, a linear determine the method. The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method.

One can distinguish between explicit and implicit methods. Iterative methods such as Newton's method are often used to solve the implicit formula.

. That value is then used in an implicit formula to "correct" the value. The result is a predictor–corrector method.

Examples

.

One-step Euler

Euler's method can be viewed as an explicit multistep method for the degenerate case of one step.

Two-step Adams–Bashforth

Euler's method is a one-step method, so the two-step Adams–Bashforth method is more accurate than Euler's method. This is always the case if the step size is small enough.

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

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 (;

can be determined as follows. Use polynomial interpolation to find the polynomial p The Lagrange formula The polynomial p instead. This equation can be solved exactly; the solution is simply the integral of p The Adams–Bashforth method arises when the formula for p by its interpolant p incurs an error of order hs, and it follows that the s-step Adams–Bashforth method has indeed order s

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

Adams–Moulton methods

. Again the b, an s, while an s-step Adams–Bashforth methods has only order s.

The Adams–Moulton methods with s = 0, 1, 2, 3, 4 are (; ) listed, where the first two methods are the backward Euler method and the trapezoidal rule (also known as the Crank-Nicolson method

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 ; had the same idea. Adams used Newton's method to solve the implicit equation .

Backward differentiation formulas (BDF)

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 equationss.

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

? More precisely, a multistep method is consistent if the local truncation error goes to zero faster than the step size h as h goes to zero, where the local truncation error. A computation using Taylor series All the methods mentioned above are consistent .

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 as h The s-step Adams–Bashforth method has order s, while the s .

These conditions are often formulated using the characteristic polynomials In terms of these polynomials, the above condition for the method to have order p.

Stability and convergence

, but the numerical solution of an s-step method depend on all the s. 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 .

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 .

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 .

Furthermore, if the method is convergent, the method is said to be strongly stable 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 This polynomial is called the stability polynomial of the multistep method. If all of its roots have modulus less than one then the numerical solution of the multistep method will converge to zero and the multistep method is said to be absolutely stable for that value of hλ. The method is said to be A-stable if it is absolutely stable for all hλ with negative real part. The region of absolute stability is the set of all hλ for which the multistep method is absolutely stable . For more details, see the section on stiff equations and multistep methods.

Example

Consider the Adams–Bashforth three-step method is the only root of modulus 1, the method is strongly stable.

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 and the second in .

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 .

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 .

See also

• Digital energy gain

References

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External links