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Optimal subsets in the stability regions of multistep methods

Lajos Lóczi

TL;DR

This work develops an exact, algebraic framework for stability-region optimization of linear multistep and multiderivative multistep methods. By transforming root-locus boundary conditions into implicit algebraic curves via Weierstrass substitution and Schur–Cohn reductions, it yields exact stability angles for the BDF and Enright families and exact stability radii for BDF, while also solving shape-optimization problems in parametric IMEX families. The authors demonstrate two key optimality results: in a two-parameter IMEX family, the maximal infinite sector has angle $\alpha=\arctan(1/2)$ attained at $(\beta_1,\beta_0)=(3/8,3/4)$; and for the same family, the largest inscribed parabola corresponds to $m=6/5$ at $(\beta_1,\beta_0)=(1/5,37/40)$, with a precise touching point on the boundary. These results provide exact, practically relevant metrics for stability guarantees and guide the design of stable multistep schemes, including IMEX methods, without relying on numerical root-locus plots.

Abstract

In this work we study the stability regions of linear multistep or multiderivative multistep methods for initial-value problems by using techniques that are straightforward to implement in modern computer algebra systems. In many applications, one is interested in (i) checking whether a given subset of the complex plane (e.g. a sector, disk, or parabola) is included in the stability region of the numerical method, (ii) finding the largest subset of a certain shape contained in the stability region of a given method, or (iii) finding the numerical method in a parametric family of multistep methods whose stability region contains the largest subset of a given shape. First we describe a simple procedure to exactly calculate the stability angle $α$ in the definition of $A(α)$-stability. As an illustration, we consider two finite families of implicit multistep methods: we exactly compute the stability angles for the $k$-step BDF methods ($3\le k\le 6$) and for the $k$-step second-derivative multistep methods of Enright ($3\le k\le 7$). Next we determine the exact value of the stability radius in the BDF family for each $3\le k\le 6$, that is, the radius of the largest disk in the left half of the complex plane, symmetric with respect to the real axis, touching the imaginary axis and lying in the stability region of the corresponding method. Finally, we demonstrate how some Schur--Cohn-type theorems of recursive nature and not relying on the RLC method can be used to exactly solve some optimization problems within infinite parametric families of multistep methods. As an example, we choose a two-parameter family of implicit-explicit (IMEX) methods: we identify the unique method having the largest stability angle in the family, then we find the unique method in the same family whose stability region contains the largest parabola.

Optimal subsets in the stability regions of multistep methods

TL;DR

This work develops an exact, algebraic framework for stability-region optimization of linear multistep and multiderivative multistep methods. By transforming root-locus boundary conditions into implicit algebraic curves via Weierstrass substitution and Schur–Cohn reductions, it yields exact stability angles for the BDF and Enright families and exact stability radii for BDF, while also solving shape-optimization problems in parametric IMEX families. The authors demonstrate two key optimality results: in a two-parameter IMEX family, the maximal infinite sector has angle attained at ; and for the same family, the largest inscribed parabola corresponds to at , with a precise touching point on the boundary. These results provide exact, practically relevant metrics for stability guarantees and guide the design of stable multistep schemes, including IMEX methods, without relying on numerical root-locus plots.

Abstract

In this work we study the stability regions of linear multistep or multiderivative multistep methods for initial-value problems by using techniques that are straightforward to implement in modern computer algebra systems. In many applications, one is interested in (i) checking whether a given subset of the complex plane (e.g. a sector, disk, or parabola) is included in the stability region of the numerical method, (ii) finding the largest subset of a certain shape contained in the stability region of a given method, or (iii) finding the numerical method in a parametric family of multistep methods whose stability region contains the largest subset of a given shape. First we describe a simple procedure to exactly calculate the stability angle in the definition of -stability. As an illustration, we consider two finite families of implicit multistep methods: we exactly compute the stability angles for the -step BDF methods () and for the -step second-derivative multistep methods of Enright (). Next we determine the exact value of the stability radius in the BDF family for each , that is, the radius of the largest disk in the left half of the complex plane, symmetric with respect to the real axis, touching the imaginary axis and lying in the stability region of the corresponding method. Finally, we demonstrate how some Schur--Cohn-type theorems of recursive nature and not relying on the RLC method can be used to exactly solve some optimization problems within infinite parametric families of multistep methods. As an example, we choose a two-parameter family of implicit-explicit (IMEX) methods: we identify the unique method having the largest stability angle in the family, then we find the unique method in the same family whose stability region contains the largest parabola.

Paper Structure

This paper contains 25 sections, 6 theorems, 134 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Structure of the paper
  4. Preliminaries
  5. Notation
  6. The Schur--Cohn reduction
  7. The stability region of a multistep method
  8. The RLC of a LMM
  9. RLCs for the BDF methods
  10. The RLC of a multiderivative multistep method
  11. RLCs for the Enright methods
  12. Optimal sector inclusions
  13. The RLC in implicit algebraic form
  14. Results for the BDF methods
  15. Results for the Enright methods
  16. ...and 10 more sections

Key Result

Theorem 2.3

$Q\in\mathbf{Sch} \Leftrightarrow (|\mathfrak{l c}\, Q|>|\mathfrak{c c}\, Q| \text{ and } Q^\mathbf{r}\in\mathbf{Sch})$.

Figures (13)

  • Figure 1: RLCs for the $k$-step BDF methods for $1\le k\le 6$. The stability region of the method in each case is the unbounded component of ${\mathbb{C}}$.
  • Figure 2: RLC for the unstable $7$-step BDF method in red (left), and a close-up near the origin (right). For comparison, the curves from Figure \ref{['fig:1']} are also superimposed as dashed gray curves.
  • Figure 3: The black curve in the left figure shows the boundary $\partial {\mathcal{S}}$ of the stability region of the (unstable) $7$-step BDF method; $\partial{\mathcal{S}}$ is non-differentiable at one point. The stability region is the unbounded outer component. The red curve segment near the origin is not part of $\partial {\mathcal{S}}$, it is a subset only of the RLC as displayed in Figure \ref{['splitfigure']}. The small brown rectangle in the center is shown in detail in the right figure. The red curve in the right figure is again the RLC. The 6 black dots depict the set of $\mu$ values such that $P_1(\cdot,\mu)$ in \ref{['P1def']} has multiple roots (there are no other $\mu\in\mathbb{C}$ parameters with this property for $k=7$). The polynomial $P_1(\cdot,\mu)$ has 1, 2 and 3 roots outside the unit disk for $\mu$ values in the dark brown, light brown and orange regions, respectively; $P_1(\cdot,\mu)$ cannot have 4 or more roots outside the unit disk. Each of the three self-intersections of the RLC in this figure (as well as the self-intersection of the RLC seen only in the left figure) corresponds to a $\mu$ value for which $P_1(\cdot,\mu)$ has two distinct roots with modulus 1. Exactly computing, for example, the unique value of $\mu_\dagger\approx -2.68886\cdot 10^{-6}+ 0.275988 i$ in the open upper half-plane where the RLC crosses itself was a non-trivial task: it took Mathematica 86 minutes to explicitly determine the coefficients of the integer polynomial defining $\mu_\dagger$ and having degree 30. The RLCs for the $k$-step BDF methods with $1\le k \le 6$ do not have any self-intersections; other singularities may occur, see Figure \ref{['BDF_6_cusp']}.
  • Figure 4: RLCs for the $k$-step Enright methods for $1\le k\le 7$. The stability region of the method in each case is the unbounded component of ${\mathbb{C}}$.
  • Figure 5: RLCs for the unstable $8$-step Enright method in red. The stability region ${\mathcal{S}}$ is not connected, ${\mathbb{C}}\setminus{\mathcal{S}}$ is the annulus-like region. For comparison, the curves from Figure \ref{['fig:3']} are displayed as dashed gray curves.
  • ...and 8 more figures

Theorems & Definitions (33)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • ...and 23 more