A collocation formulation of multistep methods for variable step-size extensions

A collocation formulation of multistep methods forvariable step-size extensions

Carmen Arevalo a,1 , Claus Fuhrer b,2 , Monica Selva c

aDepartment of Scientific Computing and Statistics, Simon Bolıvar University,Apartado 89000, Caracas 1080-A, Venezuela ([email protected])

bNumerical Analysis, Centre for Mathematical Sciences, Lund University, Box118, S-221 00 Lund, Sweden ([email protected]).

cCenter for Statistics and Mathematical Software (CESMA), Simon BolıvarUniversity, Apartado 89000, Caracas 1080-A, Venezuela ([email protected])

Abstract

Multistep methods are classically constructed by specially designed differenceoperators on an equidistant time grid. To make them practically useful, they have tobe implemented by varying the step-size according to some error-control algorithm.It is well known how to extend Adams and BDF formulas to a variable step-sizeformulation. In this paper we present a collocation approach to construct variablestep-size formulas. We make use of piecewise polynomials to show that every k-stepmethod of order k + 1 has a variable step-size polynomial collocation formulation.

Key words: ordinary differential equations (ODE), multistep methods, variablestep-size formulas, collocation.

1 Introduction

Ordinary differential equations with initial values,

x = f(t, x), x(t0) = x0, t ∈ [t0, tF ] (1.1)

1 Partially funded by Simon Bolıvar University Project DID–S1–CB–118,CONICIT contract G-97-000592 and TFR “Visiting Professor Programme; Womenin the Engineering Sciences” contract 288–99–900.2 Partially funded by the Swedish Research Council for Engineering Sciences (TFR)contract 222–96–520.

Preprint submitted to Elsevier Preprint

can be solved by linear multistep methods, defined by means of the differenceequation

xn = −k∑

i=1

αk−i,n xn−i + hn

k∑i=0

βk−i,n f(tn−i, xn−i), (1.2)

where tn = tn−1 + hn. The coefficients of the method, αk−i,n, i = 1 : k, andβk−i,n, i = 0:k, depend on ratios of the last k step-sizes,

ri =hi

hi−1

, i = n− k + 1:n. (1.3)

The development of implicit multistep methods started by considering the dif-ferential equation in its integral form, and replacing f(t, x(t)) by an interpo-lating polynomial p passing through the points (ti, f(ti, xi)), i = n, . . . , n− k,so that the k-step formula becomes

xn = xn−1 +∫ tn

tn−1

p(t) dt. (1.4)

The formulas thus obtained are the variable step-size Adams methods. Othersimilar methods can be derived by numerical integration.

A different approach to constructing variable step-size multistep formulas is touse numerical differentiation. In this case, the interpolating polynomial mustsatisfy the differential equation, p(tn) = f(tn, p(tn)). The formulas obtainedin this fashion are the backward differentiation formulas (BDF).

We are interested in devising a systematic approach to constructing variablestep-size multistep formulas of high order of accuracy, and because stable k-step explicit formulas can in general only attain order k, we restrict our studyto k-step implicit formulas of order k+1. The well-known Adams–Moulton for-mulas belong to this class, as well as the implicit difference-corrected (IDC)formulas developed for the solution of differential-algebraic equations of in-dex 2 [1].

While there are several working codes based on Adams–Moulton formulas, noimplementations are available for most of the implicit multistep formulas oforder k + 1, including the IDC formulas. To construct a solver based on aparticular formula one must have it in variable step-size form. As opposed tothe Adams–Moulton methods, most other formulas are not defined in a waythat leads directly to variable step-size formulations and thus such extensionsare not straightforward or unique.

The easiest way to implement step-size changes is to use the Nordsieck repre-sentation of a multistep method [5]. This is the tool well-known codes such asVODE [2] use.

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2 Multistep methods in the Nordsieck formulation

Consider a fixed step-size k-step linear multistep formula of order at least k,

k∑i=0

αk−i xn−i = hk∑

i=0

βk−i xn−i (2.1)

The generating polynomials of the formula, defined as

ρ(ζ) =k∑

i=0

αi ζi and σ(ζ) =

k∑i=0

βi ζi (2.2)

and normalized by αk = 1, may be written as

ρ(ζ) = det(ζI − P ) · eT2 (ζI − P )−1l (2.3a)

σ(ζ) = det(ζI − P ) · eT1 (ζI − P )−1l (2.3b)

where l = (l1, l2, . . . , lk+1) with l2 = 1 is a unique vector that defines themethod, P is the Pascal triangle matrix, e1 = (1, 0, 0, . . . , 0)T and e2 =(0, 1, 0, . . . , 0)T [4, pp. 412–417]. The k-step Nordsieck vector is

yn = (xn, hxn,h2

2!xn, . . . ,

hk

k!x(k)

n )T (2.4)

and for a single equation, without loss of generality, the method may be writtenas

yn = Pyn−1 + l (hf(tn, xn)− eT2 Pyn−1). (2.5)

The Nordsieck formulation of a multistep method relates its fixed step-sizeformula to a polynomial of degree k, given in terms of the Nordsieck vector,

p(t) = xn + xn(t− tn) + · · ·+ x(k)n

k!(t− tn)k. (2.6)

Although the Nordsieck representation is developed for fixed step-size for-mulas, this formulation can be extended to the variable step-size case forAdams-Moulton and BDF methods by constructing the collocation polyno-mial that defines the particular method. In view of this, in order to developvariable step-size formulas for general multistep methods, we look into theirrepresentation as collocation methods.

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3 The collocation approach

In the context of partial differential equations there are two main ways ofdiscretizing the continuous problem. The difference method approach consistsin the substitution of divided differences for derivatives. In the finite elementapproach the infinite-dimensional set of admissible functions is substituted bya finite-dimensional subset consisting of piecewise polynomials. Inspired bythis theory and by Butcher [3, p. 230], we formalize the definition of collocationmethods and reformulate the initial value ODE within that context. Thisapproach will allow multistep methods to be formulated directly in a variablestep-size form.

3.1 Collocation methods

Definition 1 Let V be an affine function space, I ⊂ R, and φ : V → C0(I).Consider the problem of finding v ∈ V such that

φ(v)(t) = 0 ∀t ∈ I. (3.1)

Let Vs be an s-dimensional affine subspace of V. A collocation method forsolving (3.1) finds an approximation v∗ ∈ Vs to v by requiring that

φ(v∗)(t) = 0 ∀t ∈ S ⊂ I, where S is a nonempty finite set.

The elements of S are the collocation points and v∗ is a collocation solutionof (3.1).

In order to discuss collocation methods for ODEs, we describe the initialvalue problem (1.1) in the form (3.1), by setting φ(x)(t) = x(t) − f(t, x)and I = [t0, tF ]. The affine space V can be described in terms of a particularsolution of the ODE and the vector space of C1 functions in I that satisfy thehomogeneous equation, i.e., V = xP + V0, where xP (t0) = x0 and V0 = {x ∈C1(I) | x(t0) = 0}.

Take Vs = xP +V0s , where V0

s , called the collocation space, is an s-dimensionalsubspace of V0 that will be specified later by interpolation conditions. A col-location method for solving this problem finds an approximation x∗ ∈ Vs to xsuch that

φ(x∗)(t) = x∗(t)− f(t, x∗(t)) = 0 ∀t ∈ S ⊂ I.

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These conditions will be called collocation conditions. When applying a mul-tistep method to solve the differential equation, at each time-step we need toconsider I = [tn−1, tn], and the “initial” condition can be stated as xP (tn−1) =xn−1.

3.2 Polynomial collocation methods

When Vs is a subset of Pm, the space of polynomials of degree ≤ m, withp(tn−1) = xn−1, V0

s = {q ∈ Pm | q(tn−1) = 0}, the collocation method definedabove will be called a polynomial collocation method for solving explicit ODEs.

It is known that certain, but not all, implicit Runge–Kutta methods are poly-nomial collocation methods. In the case of multistep methods, a more generalresult can be shown.

Lemma 2 Consider a multistep method of order at least k satisfying (2.3) andlet p be the polynomial defined in (2.6). If eT

1 P−1l = 0 the initial condition inI = [tn−1, tn] is satisfied. If additionally l2 = 1, the method is a polynomialcollocation method defined by xn = p(tn). Furthermore, if eT

2 P−1l = 0, then pinterpolates xn−1 at tn−1.

PROOF. Note that (P−1)ij = (−1)i+jPij. Multiplying (2.5) by eT1 P−1, with

eT1 P−1l = 0 we get p(tn−1) = xn−1. Multiplying (2.5) by eT

2 and using l2 = 1we get p(tn) = f(tn, xn), and as the approximation to x(tn) is xn = p(tn), thisis a collocation method with tn as a collocation point. Multiplying (2.5) byeT2 P−1, with eT

2 P−1l = 0 we obtain p(tn−1) = xn−1. �

By considering the Nordsieck formulation of multistep methods, it is possibleto identify an important class of multistep methods as collocation methods.

Lemma 3 Every k-step multistep method of order k + 1 is a polynomial col-location method with interpolation condition p(tn−1) = xn−1.

PROOF. Let (ρ, σ) be the generating polynomials of the method. Defineρ(ζ) = ζρ(ζ) and σ(ζ) = ζσ(ζ), so that the method (ρ, σ) is a (k + 1)-stepmethod reducible to (ρ, σ) and is also order k + 1. There is a unique vectorl = (l1, l2, l3, . . . , lk+2) (with l2 = 1) that defines the method, satisfying (2.3).It can also be shown that l2 = 1 implies that αk+1 = αk = 1. From (2.3a)we get that ρ(0) = (−1)k+1eT

2 P−1l, so eT2 P−1l = 0. From (2.3b) we obtain

σ(0) = (−1)k+1eT1 P−1l = 0. Thus, by Lemma 2 the Nordsieck polynomial for

(ρ, σ) is a collocation polynomial satisfying p(tn−1) = xn−1. �

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A significant feature of the definition of collocation methods is that fixed step-sizes are not required. Variable step-size backward differentiation (BDF) andAdams–Moulton (AM) methods can be formulated as collocation methods ina straightforward way.

Example 4 (BDFs and AM formulas as collocation methods)Consider the problem of finding a solution to the initial value problem (1.1)in the interval [tn−1, tn].

Define tn as the single collocation point.

For the k-step BDF, let p be a polynomial with the properties

p(tn−i) = xn−i, i = 1:k.

The collocation space is defined by V1 = p + P0k , where

P0k = {q ∈ Pk | q(tn−i) = 0, i = 1:k}.

For the k-step Adams–Moulton method,

p(tn−1) = xn−1 and p(tn−i) = xn−i, i = 1:k,

and the collocation space is defined by V1 = p + P0k+1, where

P0k+1 = {q ∈ Pk+1 | q(tn−1) = 0 and q(tn−i) = 0, i = 1:k}.

For general multistep methods it is not apparent how the conditions at thenodes should be defined in order to transform any formula into a collocationmethod. Moreover, we would like to generate variable step-size formulas ascollocation methods without having to construct the corresponding fixed step-size formulas, but for each particular method the conditions at the nodes maydepend in general on the coefficients of its fixed step-size formula.

3.3 Polynomial collocation and the modifier polynomial

For multistep methods the collocation space V01 is of dimension one and thus

the collocation polynomial can be written as

p∗(t) = p(t) + γp0(t), (3.2)

where the parameter γ ∈ R is determined by the collocation condition.

A case of particular interest is when p is chosen to be the collocation poly-nomial of the previous step. This case is investigated by Skeel [7] in order to

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formulate the collocation polynomial in terms of αi and βi, the coefficients of afixed step-size method. This formulation leads to expressions for αi,n and βi,n,the coefficients of a variable step-size extension of the multistep formula. Theelement p0 ∈ Pk+1 in (3.2) is called the modifier polynomial, and it dependsonly on the particular method coefficients and not on the previously computedsolution values or on the differential equation to be solved.

In the fixed step-size case, in order to specify the collocation space V01 by

interpolation conditions we define the following operators on Pk+1,

Ljh p(t) :=

j∑i=0

−αj−i p(t− ih) + hβj−i p(t− ih), j = 0:k − 1, (3.3)

and set formally

Ljh xn :=

j∑i=0

−αj−i xn−i + hβj−i xn−i, j = 0:k − 1. (3.4)

Skeel [6, p. 1235] shows that if (ρ, σ) is a linear k-step formula of order atleast k +1 with no common factors, then p0 is uniquely given by the followinginterpolation conditions,

p0(tn) = βk (3.5a)

h p0(tn) = αk (3.5b)

p0(tn−1) = 0 (3.5c)

p0(tn−1) = 0 (3.5d)

Ljh p0(tn−2) = 0, j = 0:k − 3. (3.5e)

These conditions define the collocation space V01 , of dimension one. Further-

more, p ∈ Pk+1 is given by the k + 2 interpolation conditions

p(tn−j−1) = xn−j−1, j = 0, 1, (3.6a)

p(tn−j−1) = xn−j−1, j = 0, 1, (3.6b)

Ljh p(tn−3) =Lj

h xn−3, j = 0:k − 3. (3.6c)

This way of formulating the collocation polynomial has the drawback thatthe interpolation conditions are given in terms of the coefficients αi, βi. Asin the variable step-size case these coefficients are unknown, an extension tothat case is not straightforward. Then the coefficients will vary from step tostep, depending on the last k step-sizes. Denoting the coefficients of a variable

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step-size method by αi,n and βi,n, Skeel extends the definition of Ljh to

Ljhn,np(t) :=

j∑i=0

−αj−i,n p(t−ihn)+hn βj−i,n p(t−ihn), j = 0:k−1. (3.7)

Then the coefficients αi,n and βi,n are determined by requiring

Ljhn,n p0(tn − 2hn) := 0, j = 0:k − 2. (3.8)

Note that p0 is based on the fixed step-size formulation, i.e., it is the polyno-mial defined by (3.5). Condition (3.8) can be viewed as an extension of (3.5e)inasmuch as both are the same in the case of fixed step-sizes.

Equations (3.8) establish conditions for the coefficients of the variable step-size method. These coefficients can be uniquely defined from the equations to-gether with k+2 order conditions and the normalization condition αk,n := αk.By requiring (3.8), Skeel’s construction builds the variable step-size coeffi-cients starting from a fixed step-size formula. In [7] it is shown that in thecase of Adams–Moulton methods the standard variable step-size extension isobtained.

In the next section condition (3.8) will be replaced by geometrical conditionsthat will enable us to define collocation spaces based purely on solution dataand not in terms of method coefficients.

4 Piecewise polynomial collocation

Instead of defining a collocation method by means of a single polynomial,we will consider piecewise polynomials for that purpose. The idea behind thepiecewise polynomial collocation formulation of a method is to construct acollocation polynomial as described in Section 3.1, linking past data to thenew approximations xn and xn in such a way as to cover the whole spectrumof linear multistep methods. As we are interested in k-step methods of orderk + 1, we will construct a piecewise polynomial of degree k + 1,

p(t) := pi(t) for t ∈ [tn−i−1, tn−i], i = 0:k − 1 (4.1a)

satisfying the interpolation condition of Lemma 3, and take the extension ofp0 over R as the collocation polynomial with collocation at tn that defines themethod, i.e., xn = p0(tn) and xn(tn) = p0(tn).

While we will not demand that p0 interpolate past values at the last k points,each polynomial pi will link p0 to past data by interpolation conditions,

pi(tn−i−1) = xn−i−1 and pi(tn−i−1) = xn−i−1, i = 1:k − 1 (4.1b)

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and the collocation polynomial itself will interpolate the last computed values,

p0(tn−1) = xn−1 and p0(tn−1) = xn−1. (4.1c)

The information on past values will be transmitted to the collocation polyno-mial by requiring this polynomial to “connect” to each polynomial piece in asmooth way,

p(j)0 (tn−i) = p

(j)i (tn−i), j = 0:k − 1, i = 1:k − 1 (4.1d)

(see Figure 1.)

Fig. 1. The piecewise collocation polynomial for the 4-step method IDC34. Thecollocation polynomial p0 is extended over [tn−4, tn].

Constructing the piecewise polynomial in this manner still allows for k − 1degrees of freedom. Thus, for each i = 1:k − 1 we will require

p(k)0 (τi) = p

(k)i (τi) (4.1e)

where τi ∈ R are method-related parameters, or alternatively, that

p(k+1)0 = p

(k+1)i . (4.1f)

This last condition allows the slopes of the straight lines y = p(k)0 (t) and

y = p(k)i (t) to be the same. Then either condition (4.1e) is trivially satisfied

or else the lines are parallel and can be considered to intersect at τi = ∞.Although only (4.1f) covers the latter situation, condition (4.1e) also coversthe first case with arbitrary τi.

The collocation polynomial method defined in this manner has order k +1, asis shown in the following theorem.

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Theorem 5 Every k-step piecewise polynomial collocation formulation yieldsa k-step variable step-size formula of order at least k + 1.

PROOF. Let us suppose a piecewise polynomial collocation is used to solvethe initial value problem

x = f(t, x), t ∈ [tn−1, tn], x(tn−1) = xn−1, (4.2)

whose exact solution is a polynomial x(t) of degree ≤ k + 1.

Denote by p(t) the piecewise polynomial defined in 4.1,

p(t) = pi(t) t ∈ [tn−i−1, tn−i], i = 0:k − 1,

and define Λ(t) = p(t)−x(t). This is a piecewise polynomial of degree ≤ k+1,

Λi(t) = pi(t)− x(t) t ∈ [tn−i−1, tn−i], i = 0:k − 1 (4.3)

that satisfies the following conditions

Λi(tn−i−1) = 0, i = 0:k − 1, (4.4a)

Λi(tn−i−1) = 0, i = 0:k − 1, (4.4b)

Λ(j)i (tn−i)− Λ

(j)0 (tn−i) = 0 for j = 0:k − 1, i = 1:k − 1, (4.4c)

Λ(k)i (τi)− Λ

(k)0 (τi) = 0, i = 1:k − 1, (4.4d)

Λ0(tn) = 0. (4.4e)

When condition (4.1f) is used instead of (4.1e),

Λ(k+1)i − Λ

(k+1)0 = 0 (4.4f)

replaces (4.4d) for the corresponding i. The matrix of this linear system ofequations for the coefficients of Λ is identical to the matrix of the systemthat defines the piecewise polynomial p. If this polynomial is well defined,the piecewise polynomial Λ is identical to 0, and thus p ≡ x, so the k-stepvariable step-size formula we obtain setting xn = p(tn) is exact and thus hasorder k + 1. �

Example 6 (2-step methods as polynomial collocation methods)Consider the problem of finding a solution to the initial value problem (4.2).

Let V1 be the set of all piecewise cubic polynomials of the form

p(t) =

p1(t) ∈ P3 , t ∈ [tn−2, tn−1]

p0(t) ∈ P3 , t ∈ [tn−1, tn](4.5)

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where the polynomials p0 and p1 are defined by the following conditions:

p1(tn−2) = xn−2 p1(tn−1) = p0(tn−1) p1(τ1) = p0(τ1)p1(tn−2) = xn−2 p1(tn−1) = p0(tn−1)p0(tn−1) = xn−1

p0(tn−1) = xn−1

and where τ1 is a fixed, method-dependent point. An approximation p∗ ∈ V1 tothe exact solution is found by setting

p0(tn) = xn.

Letting xn = p∗(tn) we get a variable step-size formula for xn,

xn = −3r3n − 6η1r

3n + 3rn − 6η1 + 6

6η1 − 3rn − 6xn−1 −

r3n(6η1 − 3)

6η1 − 3rn − 6xn−2

+hn

(3η1 − rn − 3

6η1 − 3rn − 6xn −

3r2nη1 − r2

n + 2rn − 3η1 + 3

6η1 − 3rn − 6xn−1

− r2n(3η1 − 2)

6η1 − 3rn − 6xn−2

)(4.6)

where rn = hn/hn−1, η1 = (τ1 − tn−2)/hn−1. For each value of the parameterwith τ1 6= (tn−1+tn)/2, this is a variable step-size 2-step formula. As η1 varies,a continuum of 2-step formulas of order 3 is generated; for η1 = ∞ (4.6) isreducible to the trapezoid formula, a 1-step method of order 2.

When η1 = 1/2 (i.e., τ1 = (tn−2 + tn−1)/2), this formula is identical to the onepresented in [4] for the variable stepsize Adams-Moulton methods, as obtainedfrom (1.4).

4.1 Properties of the polynomial collocation methods

Lemma 7 Any k-step polynomial collocation method with p(tn−1) = xn−1 canbe written in a k-step piecewise polynomial collocation formulation.

PROOF. Let p∗ ∈ Pk+1 be the collocation polynomial defined in Section 3,such that the formula defined by xn = p∗(tn), is a k-step method. We want tofind a piecewise polynomial p = pi, i = 1 : k − 1, as in Section 4, satisfyingall the interpolation, smoothness and parametric conditions. Let p0 ≡ p∗ and

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define di = pi − p0 ∈ Pk+1. Then di, i = 1 : k − 1, is a polynomial of degreek + 1 satisfying the k + 2 conditions

di(tn−i−1) = xn−i−1 − p∗(tn−i−1), (4.7a)

di(tn−i−1) = xn−i−1 − p∗(tn−i−1), (4.7b)

d(j)i (tn−i) = 0, j = 0:k − 1. (4.7c)

Thus, di(t) = ai,k+1(t− tn−i)k+1 + ai,k(t− tn−i)

k with

ai,k+1 =(−1)k

hk+1n−i

(k di(tn−i−1) + hn−i di(tn−i−1)

), (4.8a)

ai,k =(−1)k

hkn−i

((k + 1) di(tn−i−1) + hn−idi(tn−i−1)

). (4.8b)

Suppose there is an i such that condition (4.1e) is not satisfied for any τi ∈ R.

Then y = d(k)i (t) is a straight line with zero slope, i.e., ai,k+1 = 0. But then di

is of degree k and thus condition (4.1f) is satisfied. �

Corollary 8 Every polynomial collocation method with p(tn−1) = xn−1 hasorder k + 1.

Theorem 9 Every implicit variable step-size k-step method of order k+1 hasa piecewise polynomial collocation formulation.

PROOF. This follows from Lemmas 3 and 7. �

Theorems 5 and 9 show that with the piecewise polynomial collocation ap-proach a variable step-size extension of every k-step formula of order k + 1may be constructed by way of a collocation polynomial. Furthermore, varyingparameters τi in (−∞,∞ ] will produce an extension of every formula of orderk + 1.

The error constants of the family of k-step formulas of order k + 1 dependcontinuously on the parameters of the formula. This information might beuseful for choosing a method in a particular setting. In Figure 2 we observethe level curves for the error constants of 3-step methods in relation to the twomethod-related parameters η1 and η2. The error constants are smaller in thedarker regions of the plot. The shaded areas correspond to all stable methods.

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Fig. 2. Error constants of 3-step methods as functions of their two parameters.

4.2 The collocation polynomial of a k-step, order k + 1 method

If ai,k+1 6= 0 in (4.8a) for every i = 1 : k − 1, the collocation polynomial ofdegree k + 1 that defines a k + 1 order multistep method can be described bythe k + 2 conditions

p∗(tn) = xn, (4.9a)

p∗(tn−1) = xn−1, (4.9b)

p∗(tn−1) = xn−1, (4.9c)

(k + 1) ai,k+1 (τi − tn−i) + ai,k = 0, i = 1:k − 1. (4.9d)

Substituting (4.8) in (4.9d) and setting ηi = (τi − tn−i−1)/hn−i we get

(ηi −k − 1

k) p∗(tn−i−1) +

1

k(ηi −

k

k + 1) hn−i p

∗(tn−i−1) =

(ηi −k − 1

k) xn−i−1 +

1

k(ηi −

k

k + 1) hn−i xn−i−1, i = 1:k − 1, (4.9e)

which can replace (4.9d) in the above system. This set of equations now definesthe collocation polynomial as a function of the k − 1 parameters ηi.

In the case when no τi can be found to satisfy (4.9d), as happens when condi-tion (4.1f) must be used to define the collocation polynomial, the correspond-ing equation in (4.9e) must be substituted by

k p∗(tn−i−1) + hn−i p∗(tn−i−1) = k xn−i−1 + hn−i xn−i−1. (4.9f)

The collocation polynomial can thus be constructed directly in terms of the

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method parameters from (4.9), without having to build upon piecewise poly-nomials. For example, it can be seen from the geometric properties of Adams–Moulton methods (see Example 4) that they are obtained by setting all theparameters in (4.9e) equal to (k − 1)/k.

5 Summary

Our aim in this work was to formulate k-step methods of order k+1 as variablestep-size formulas. In order to accomplish this we formalized the definition ofcollocation methods for initial value ODEs. Based on this theory, we were ableto construct the numerical solution to such problems by means of piecewisepolynomials. This led to the characterization of k-step methods by k − 1parameters, smoothly related to particular properties such as error constants.We proved that a k-step variable step-size formula is of order k +1 if and onlyif it has a piecewise polynomial collocation formulation. We also showed therelation between our methods, the Nordsieck formulation and Skeel’s variablestep-size extensions.

This work provides insight in the functional formulation of variable step-sizemultistep methods. There are several possible extensions to this work. Stabil-ity of variable step-size formulas must be studied in detail, as well as errorconstants and regions of stability. All these concepts may be looked at in theirrelation to the parametric conditions that define each method. Another areaof interest is the re-use of collocation polynomials as predictors and in errorestimation. Finally, it would be of interest to see how these ideas could beused to study other families of multistep methods, in particular the implicitk-step, order k methods, which include BDFs.

Acknowledgements

The first author thanks Steve Campbell at the Department of Mathematics,North Carolina State University, and all authors thank CESMa, U.S.B., andthe Department of Numerical Analysis, L.U., for the peaceful and stimulatingenvironments they enjoyed on the several occasions these institutions hostedtheir visits.

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