Informatics Educational Institutions & Programs
Contents
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
The result is approximations for the value of at discrete times :
Multistep methods use information from the previous steps to calculate the next value. In particular, a linear multistep method uses a linear combination of and to calculate the value of for the desired current step. Thus, a linear multistep method is a method of the form
One can distinguish between explicit and implicit methods. If , then the method is called "explicit", since the formula can directly compute . If then the method is called "implicit", since the value of depends on the value of , and the equation must be solved for . 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 . 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
One-step Euler
A simple numerical method is Euler's method:
This method, applied with step size on the problem , gives the following results:
Two-step Adams–Bashforth
Euler's method is a one-step method. A simple multistep method is the two-step Adams–Bashforth method
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 and , while the 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):
The coefficients can be determined as follows. Use polynomial interpolation to find the polynomial p of degree such that
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 and . 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 , an s-step Adams–Moulton method can reach order , 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:
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 , as above, but also . The coefficients are given by
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 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
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 as h goes to zero. This is equivalent to the following condition on the coefficients of the methods:
These conditions are often formulated using the characteristic polynomials
Stability and convergence
The numerical solution of a one-step method depends on the initial condition , but the numerical solution of an s-step method depend on all the s starting values, . 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 (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 all converge to the initial value as . Then, the numerical solution converges to the exact solution as 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 (Süli & Mayers 2003, p. 340).
Furthermore, if the method is convergent, the method is said to be strongly stable if 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
Example
Consider the Adams–Bashforth three-step method
The other characteristic polynomial is
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).
See also
References
- Bashforth, Francis (1883), An Attempt to test the Theories of Capillary Action by comparing the theoretical and measured forms of drops of fluid. With an explanation of the method of integration employed in constructing the tables which give the theoretical forms of such drops, by J. C. Adams, Cambridge
{{citation}}
: CS1 maint: location missing publisher (link). - Butcher, John C. (2003), Numerical Methods for Ordinary Differential Equations, John Wiley, ISBN 978-0-471-96758-3.
- Dahlquist, Germund (1956), "Convergence and stability in the numerical integration of ordinary differential equations", Mathematica Scandinavica, 4: 33–53, doi:10.7146/math.scand.a-10454.
- Dahlquist, Germund (1963), "A special stability problem for linear multistep methods", BIT, 3: 27–43, doi:10.1007/BF01963532, ISSN 0006-3835, S2CID 120241743.
- Goldstine, Herman H. (1977), A History of Numerical Analysis from the 16th through the 19th Century, New York: Springer-Verlag, ISBN 978-0-387-90277-7.
- Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems (2nd ed.), Berlin: Springer Verlag, ISBN 978-3-540-56670-0.
- Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-60452-5.
- Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.
- Milne, W. E. (1926), "Numerical integration of ordinary differential equations", American Mathematical Monthly, 33 (9), Mathematical Association of America: 455–460, doi:10.2307/2299609, JSTOR 2299609.
- Moulton, Forest R. (1926), New methods in exterior ballistics, University of Chicago Press.
- Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2000), Matematica Numerica, Springer Verlag, ISBN 978-88-470-0077-3.
- Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0-521-00794-1.