# On the zero-stability of multistep methods on smooth nonuniform grids

## Abstract

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid \(\{t_n\}_{n=0}^N\) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., \(t_n = {\varPhi }(\tau _n)\), where \(\tau _n = n/N\) and the map \({\varPhi }\) is monotonically increasing with \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines \({\varPhi }\), and a tolerance requirement which determines *N*. Given any strongly stable multistep method, there is an \(N^*\) such that the method is zero stable for \(N>N^*\), provided that \({\varPhi }\in C^2[0,1]\). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy \(h_n/h_{n-1} = 1 + {\mathrm {O}}(N^{-1})\) as \(N\rightarrow \infty \). The results are exemplified for BDF-type methods.

## Keywords

Initial value problems Linear multistep methods BDF methods Zero stability Nonuniform grids Variable step size Convergence## Mathematics Subject Classification

65L04 65L05 65L06 65L07## 1 Introduction

*k*,

*Zero stability*is necessary for convergence, and requires that all roots of \(\rho (\zeta )=0\) lie inside or on the unit circle, with no multiple unimodular roots. Since consistent methods have \(\rho (\zeta ) = (\zeta -1)\cdot \rho _R(\zeta )\) as indicated above, zero stability is a condition on the

*extraneous operator*\(\rho _R(\zeta )\). Its zeros are referred to as the extraneous roots. Strong zero stability requires that

*all extraneous roots are strictly inside the unit circle*; this is a condition on the

*k*coefficients \(\{\gamma _j\}_{j=0}^{k-1}\).

Since the extraneous operator is void in Adams–Moulton and Adams–Basforth methods, these methods are trivially zero stable for variable steps [9, p. 407]. The most important case having a nontrivial extraneous operator is the BDF methods, known to be zero stable for \(1\le k\le 6\), cf. [5, 9, p. 381]. Some (nonstiff) method suites, such as the dcBDF and IDC methods [1], are based on the BDF \(\rho \) operator, and have the same zero stability properties for \(k\ge 2\). Other examples of nontrivial extraneous operators are the weakly stable explicit midpoint method (two-step method of order 2) and the lesser used weakly stable implicit Milne methods [9, p. 363].

Adaptive computations are of particular importance for stiff problems, as widely varying time scales call for correspondingly large variations in step size. Of the methods mentioned above, only the BDF family has unbounded stability regions specifically designed for stiff problems. Thus the BDF methods must handle step size variations well, in spite of its extraneous operator, explaining why studies of variable step size zero stability mostly center on the BDF methods [9, p. 402ff].

*grid-independent representation*of multistep methods [2]. This represents a multistep method on any nonuniform grid using a fixed parametrization, defining a computational process where the coefficients \(\alpha _{j,n}, \beta _{j,n}\) vary along the solution and depend on \(k-1\) consecutive step size ratios. For simplicity, but without loss of generality, let us consider a quadrature problem \(\dot{y} = f(t)\) on [0, 1] using variable steps. The multistep method (1.1) becomes

*y*is sufficiently differentiable, and where \(\vartheta \in [t_n,t_{n+k}]\). Subtracting (1.4) from (1.3) gives

An overview is given in [9, p. 402ff], but the classical results focus on the existence of local step size ratio bounds that guarantee zero stability. By constrast, our focus is on grid smoothness. Using (near) Toeplitz operators, our aim is to develop a proof methodology for adaptive computation, aligned with the formal convergence analysis in the Lax–Stetter framework, cf. [15]. We let the grid points be given by a strictly increasing sequence \(\{t_n\}_0^N\) and define the step sizes by \(h_n = t_{n+1}-t_n\), requiring that \(h_n\rightarrow 0\) for every *n* as \(N\rightarrow \infty \). If the grid is smooth enough, then any multistep method which is strongly zero stable on a uniform grid is also zero stable on the nonuniform grid for *N* large enough.

*N*is large enough. The important issues are to generate a smooth step size sequence (which automatically manages step size ratios), and using a sufficiently small error tolerance, which determines

*N*. Although such step size sequences can easily be constructed in adaptive computation [12], most multistep codes still use comparatively large step size changes, violating the smoothness conditions required for zero stability. This has been demonstrated to be a likely cause of poor computational stability observed in practice [13]. In production codes it is often thought that frequent, small step size changes are not “worthwhile,” but the present paper and classical theory only support such step size changes.

Our approach is intended as an analysis tool for deriving a rigorous convergence proof for adaptive multistep methods, redefining practical implementation principles. A full convergence analysis of the initial value problem \(\dot{y} = f(t,y)\) requires further attention to detail, as it also involves the Lipschitz continuity of the vector field *f* with respect to *y*, as well as (for implicit methods) the solvability of equations of the form \(v = \gamma \, hf(t,v) + w\). The solvability will depend on the magnitude of the Lipschitz constant \(L[\gamma \,hf]\) or the logarithmic Lipshitz constant \(M[\gamma \,hf]\), see e.g. [14]. Likewise, error bounds will depend on these quantities. Here, however, we only focus on zero stability, which can be fully characterized in the simpler setting of a quadrature problem. We shall return to the full convergence analysis on smooth nonuniform grids in a forthcoming study.

## 2 Smooth nonuniform grids

If an initial value problem has a smooth solution, then the step size sequence, keeping the local error (nearly) constant, is also smooth [6, 11]. A smooth sequence is also known to be necessary in connection with e.g. Hamiltonian problems [10], as well as in finite difference methods for boundary value problems. For these reasons, we shall model nonuniform grids by a smooth deformation of an equidistant grid. We only consider compact intervals.

**Adaptive computation.**The asymptotic behavior of the local error per unit step in a multistep method is of the form \(l_n = ch_n^{p}y^{(p+1)}\). The most common step size control in adaptive computation aims to keep \(\Vert l_n\Vert \) constant, equal to a given local error tolerance \(\varepsilon \). Representing the step size in terms of a

*step size modulation function*\(\mu (t)\) allows the step size at time

*t*to be expressed as \(h = \mu (t)/N\), so that the “ideal” step size sequence can be modeled by

*the local error tolerance determines*

*N*. By contrast, \(\mu (t)\) is

*determined by the problem*. It is smooth if \(y^{(p+1)}(t)\) is smooth, since

*N*, reducing step sizes as well as

*step ratios*. Thus it is justified to model a nonuniform grid by a smooth grid deformation, and such a grid can be generated using a proper filter to continually adjust the step size. It also corresponds well to the behavior observed in computational practice when such step size controllers are employed.

**Modeling a smooth nonuniform grid.**Let \({\varPhi }:\tau \mapsto t\) be a smooth, strictly increasing map in \(C^2[0,1]\), satisfying \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\). Further, let its derivative \({\varPhi }' = {\mathrm {d}}{\varPhi }/{\mathrm {d}}\tau \) be denoted by \(\varphi \) and assume that \(\varphi '/\varphi \in L^{\infty }[0,1]\). Now, given

*N*, let \(\tau _n = n/N\) and construct a smooth nonuniform grid \(\{t_n\}_{n=0}^N\) by

*step size sequence*\(\{h_n\}_{n=0}^{N-1}\) by

**Step ratios.**The coefficients of a multistep method on a nonuniform grid depend on the ratio of adjacent step sizes. By (2.2) the

*step ratios*\(\{r_n\}_{n=0}^{N-2}\) are given by

*locally the method behaves like a constant step size method*for

*N*large enough, since we assumed \(\varphi '/\varphi \in L^{\infty }[0,1]\).

**Step sizes and ratios as a function of**

*t*

**.**Using \(t={\varPhi }(\tau )\) and \({\mathrm {d}}t = \varphi \,{\mathrm {d}}\tau \), the step size modulation function \(\mu (t)\) and the derivative \(\varphi (\tau )\) satisfy the functional relation

*t*and denoting time derivatives by a dot to distinguish them from derivatives with respect to \(\tau \), we obtain \({{\dot{\mu }}} \,{\mathrm {d}}t = \varphi ' \, {\mathrm {d}}\tau \). Hence

*t*,

## 3 Deflation and operator factorization

*y*contains all successive approximations \(\{y_n\}_{n=1}^{N}\). The vector \(Y_0\) is constructed from the initial conditions, \(y_0, \dots y_{-k+1}\). Further, \(A_{N}(\varphi )\) is an \(N\times N\) matrix containing the method coefficients, and is associated with a nonuniform grid characterized by the function \(\varphi \). The step sizes are represented by a diagonal matrix \(H_N = {\tilde{\varphi }}/N\),

*N*, and an \(N^*\), such that \(\Vert A_{N}^{-1}(\varphi )H_N\Vert \le C_{\varphi }\) for all \(N > N^*\). As \(\varphi (\tau ) \equiv 1\) corresponds to a uniform grid, \(A_{N}(1)\) denotes the Toeplitz matrix of method coefficients for constant step size \(H_N = I/N\). Then zero stability is equivalent to \(\Vert A_{N}^{-1}(1)/N\Vert \le C_1\) for all

*N*.

*k*-vector \(c_n^{{\mathrm {T}}}(\varphi )\) and the backward difference operator \(\nabla = (-1\quad 1)\), i.e.,

Standard constant step size coefficients of BDF*k* methods denoted by \(a_n^{{\mathrm {T}}}(1)\), with diagonal elements of the Toeplitz operator \(A_{k,N}(1)\) in boldface. The elements appear in each row of \(A_{k,N}(1)\), to the left of (below) the diagonal

Coefficients of \(A_{k,N}(1)\) and \(R_{k,N}(1)\) for BDF2–BDF6 methods | ||||||||
---|---|---|---|---|---|---|---|---|

BDF2 | \(a_n^{{\mathrm {T}}}(1)\) | 1 / 2 | \(-2\) | 3 / 2 | ||||

\(c_n^{{\mathrm {T}}}(1)\) | \(-1/2\) | 3 / 2 | ||||||

BDF3 | \(a_n^{{\mathrm {T}}}(1)\) | \(-1/3\) | 3 / 2 | \(-3\) | 11 / 6 | |||

\(c_n^{{\mathrm {T}}}(1)\) | 1 / 3 | \(-7/6\) | 11 / 6 | |||||

BDF4 | \(a_n^{{\mathrm {T}}}(1)\) | 1 / 4 | \(-4/3\) | 3 | \(-4\) | 25 / 12 | ||

\(c_n^{{\mathrm {T}}}(1)\) | \(-1/4\) | 13 / 12 | \(-23/12\) | 25 / 12 | ||||

BDF5 | \(a_n^{{\mathrm {T}}}(1)\) | \(-1/5\) | 5 / 4 | \(-10/3\) | 5 | \(-5\) | 137 / 60 | |

\(c_n^{{\mathrm {T}}}(1)\) | 1 / 5 | \(-21/20\) | 137 / 60 | \(-163/60\) | 137 / 60 | |||

BDF6 | \(a_n^{{\mathrm {T}}}(1)\) | 1 / 6 | \(-6/5\) | 15 / 4 | \(-20/3\) | 15 / 2 | \(-6\) | 147 / 60 |

\(c_n^{{\mathrm {T}}}(1)\) | \(-1/6\) | 31 / 30 | \(-163/60\) | 79 / 20 | \(-71/20\) | 147 / 60 |

Unlike generating polynomials, the (near) Toeplitz operators have the advantage of applying also to nonuniform grids. The following factorization of \(H_N^{-1}A_{N}(\varphi )\) is then a matrix representation of the deflation operation described above.

### Theorem 3.1

*extraneous operator*, dependent on the nonuniform grid, and

*simple integrator*\({\mathbf {D}}_N^{-1}\) is stable, and for all \(N\ge 1\) it holds that \(\Vert {\mathbf {D}}_N^{-1}\Vert _{\infty } = 1\).

### Proof

*N*. \(\square \)

*diagonal dominance*. For example, by Table 1, the BDF2 matrix \(NA_{2,N}(1)\) associated with the \(\rho \) operator has the factorization

### Theorem 3.2

For every strongly stable *k*-step method on a uniform grid, there is a constant \(C_0 < \infty \), such that \(\Vert R_{k,N}^{-1}(1)\Vert _{\infty } \le C_0\) for all \(N\ge 1\).

### Proof

*u*must be absolute summable as \(N\rightarrow \infty \). By construction,

*u*satisfies the difference equation \(\rho _R({\mathrm {E}}) u = 0\), where \({\mathrm {E}}\) is the forward shift operator. By assumption \(\rho _R(\zeta )\) satisfies the strict root condition. Therefore

*u*is bounded, i.e., \(u\in l^{\infty }\). Let \(\rho _R(\zeta _{\nu })=0\) and let

## 4 Zero stability on nonuniform grids—the BDF2 method

*r*. Diagonal dominance requires that \(1+2r-r^2 > 0\), which holds if \(0< r < 1 + \sqrt{2}\), so the classical bound is obtained once more. As we assume a smooth grid in terms of (2.3), with \({\dot{\mu }} = \varphi '/\varphi \in L^{\infty }[0,1]\), the condition \(r_n < 1 + \sqrt{2}\) is fulfilled for

*u*is bounded (zero stability), then the original solution

*y*of (3.1) is obtained by simple Euler integration, \(y_{n+1} = y_n + u_n/N\), where \(h=1/N\) is a constant step size and \(N\rightarrow \infty \). Since the latter integration is stable, we only need to bound the solutions

*u*of (4.7). Using (2.4), we write the step ratios

*N*, the closer is \(|v_n|\) to zero. Now, for \(v_n\equiv 0\) we obtain the classical constant step size method. The difference equation (4.7) can then be rearranged as a Toeplitz system \(T_0 u = U_0\), where \(T_0 = R_{2,N}(1)\) and \(u = \{u_n\}_1^N\) denotes the entire solution. The vector \(U_0\) contains initial data as needed. By Theorem 3.2, we have \(\Vert T_0^{-1}\Vert _{\infty } \le C_0\) for all \(N\ge 1\).

*N*in particular, such that (4.9) is satisfied. Because \(w := \Vert V\Vert _{\infty } = {\mathrm {O}}(N^{-1})\) if the grid is regular, there is always an

*N*large enough to satisfy this condition. Considering the equation

*N*large enough to guarantee that

## 5 Zero stability on nonuniform grids—higher order methods

In a *k*-step method using variable steps, the coefficients depend on \(k-1\) step ratios. This makes the problem significantly more difficult. Without loss of generality, we will only consider an approach linear in *V* below. Note that while \(\Vert V\Vert _{\infty } = {\mathrm {O}}(N^{-1})\), it follows that higher powers of *V* are \(\Vert V\Vert _{\infty }^k = {\mathrm {O}}(N^{-k})\), implying that they are significantly smaller than the first order term when *N* is large and the grid is smooth. For example, in (4.12) above, we have \(w={\mathrm {O}}(N^{-1})\) implying that the \(w^2\) is negligible as \(N\rightarrow \infty \); it is therefore sufficient to consider terms of order \({\mathrm {O}}(N^{-1})\) only. This overcomes the added difficulty of considering *k*-step methods.

*k*-step method follows the same pattern as the in the previous examples. Neglecting quadratic and higher order terms in

*V*, the extraneous operator is

*V*, while only incurring \({\mathrm {O}}(N^{-2})\) perturbations. Further (4.11) holds for all \(V_j\).

### Theorem 5.1

For all smooth maps \({\varPhi }\) there exist constants \(N^*\) and \(C_{\varphi }\) (independent of *N*) such that \(\Vert R_{k,N}^{-1}(\varphi )\Vert _{\infty } \le C_{\varphi }\) for \(N > N^*\), whenever \(\Vert R_{k,N}^{-1}(1)\Vert _{\infty } \le C_0\) for all *N*.

*n*as indicated by (5.3). Within this setting, after deflating the operator, we obtain a recursion on a nonuniform grid corresponding to

*sufficient*conditions for zero stability,

It is important to note that we do not try to determine the greatest possible step size increase, but instead prove that every strongly stable method will be zero stable on smooth grids. We have also seen that the complexity of determining exact stability bounds quickly becomes overwhelming, which is why we argue that an alternative proof, revealing the dependence on smoothness and method parameters, is sufficient.

## 6 Conclusions

In this paper we have demonstrated that any linear multistep method which is strongly stable on a uniform grid is also zero stable on any smooth nonuniform grid. Grid smoothness is (in theory) determined by a grid map \({\varPhi }:[0,1] \rightarrow [0,1]\), satisfying \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\), and having a strictly positive derivative \(\varphi = {\varPhi }'\). The grid map transforms a uniform grid of *N* steps into a nonuniform grid, which is smooth if \(\log \varphi \) is continuously differentiable.

In practice, this corresponds to a smooth step size variation, where the step size at time \(t\in [0,1]\) can be represented by a continuous modulation function, so that \(h(t) = \mu (t)/N\). Here \({\dot{\mu }}(t) = \varphi '/\varphi \), which must remain bounded. The modulation function \(\mu (t)\) is determined by the solution of the differential equation, while *N* is determined by the accuracy requirement as specified by the tolerance \(\varepsilon \).

*k*-step method is associated with

*k*bounded Toeplitz operators \(T_0,\dots T_{k-1}\), where \(T_0\) is associated with the constant step size method. If that method is strongly zero stable, then \(T_0\) has a bounded inverse. Smooth step size variation is characterized locally by the function \(\varphi '/\varphi \), the magnitude of which determines how many steps

*N*that need to be taken in order to guarantee variable step size zero stability. Thus, if

This result is also practically significant as it implies that time step adaptivity must be implemented using smooth step size changes, such that consecutive step ratios are \(r = 1 + {\mathrm {O}}(h)\). This can easily be achieved, as there is a wide range of smooth controllers available for dedicated purposes [12]. These are based on digital filter theory, and control \(\log h\) in small increments, changing the step size on every step. Since \(h \sim \varphi /N\), such a controller keeps \(\log \varphi \) smooth, in line with the assumptions of Theorem 5.1. The smoothness requirement is local, and does not imply any bound on \(h_{\mathrm {max}}/h_{\mathrm {min}}\). It is therefore not a limitation in stiff computation, where overall step size variation necessarily is large.

## Notes

### Acknowledgements

The authors gratefully acknowledge the contribution of Prof. Carmen Arévalo, who provided the grid-independent variable step size coefficients for the BDF3 method, computed in Maple.

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