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An insight on some properties of high order nonstandard linear multistep methods

Bálint Takács

TL;DR

The paper develops high-order nonstandard linear multistep methods for autonomous ODEs, proving that, under a nonstandard Taylor expansion, the same convergence order as standard methods can be achieved when φ satisfies a precise condition. It further shows that boundedness, a monotonicity-like property, and a linear invariant are preserved for all step sizes, provided φ is bounded by a forward-Euler stability bound and starting values respect the invariants. The authors propose several φ choices to attain various orders (up to 4 in demonstrated cases) and validate the theory with numerical experiments on a logistic growth equation and an SEIR model, highlighting practical benefits and trade-offs against standard nonstandard Runge–Kutta schemes. This work offers a rigorous framework for constructing high-order, property-preserving nonstandard multistep methods with potential applications in population dynamics and epidemiological modeling, especially where large time steps are desirable without sacrificing qualitative fidelity.

Abstract

In this paper, nonstandard multistep methods are considered. It is shown that under some (sufficient and necessary) conditions, these methods attain the same order as their standard counterparts - to prove this statement, a nonstandard version of Taylor's series is constructed. The preservation of some qualitative properties (boundedness, the linear combination of the components, and a property similar to monotonicity) is also proven for all step sizes. The methods are applied to a one-dimensional equation and a system of equations, in which the numerical experiments confirm the theoretical results.

An insight on some properties of high order nonstandard linear multistep methods

TL;DR

The paper develops high-order nonstandard linear multistep methods for autonomous ODEs, proving that, under a nonstandard Taylor expansion, the same convergence order as standard methods can be achieved when φ satisfies a precise condition. It further shows that boundedness, a monotonicity-like property, and a linear invariant are preserved for all step sizes, provided φ is bounded by a forward-Euler stability bound and starting values respect the invariants. The authors propose several φ choices to attain various orders (up to 4 in demonstrated cases) and validate the theory with numerical experiments on a logistic growth equation and an SEIR model, highlighting practical benefits and trade-offs against standard nonstandard Runge–Kutta schemes. This work offers a rigorous framework for constructing high-order, property-preserving nonstandard multistep methods with potential applications in population dynamics and epidemiological modeling, especially where large time steps are desirable without sacrificing qualitative fidelity.

Abstract

In this paper, nonstandard multistep methods are considered. It is shown that under some (sufficient and necessary) conditions, these methods attain the same order as their standard counterparts - to prove this statement, a nonstandard version of Taylor's series is constructed. The preservation of some qualitative properties (boundedness, the linear combination of the components, and a property similar to monotonicity) is also proven for all step sizes. The methods are applied to a one-dimensional equation and a system of equations, in which the numerical experiments confirm the theoretical results.

Paper Structure

This paper contains 15 sections, 8 theorems, 29 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. The order of nonstandard multistep methods
  5. Preservation of qualitative properties
  6. The choice of function phi
  7. Proofs of the statements of Section \ref{['sec:main']}
  8. Applications and numerical simulations
  9. One-dimensional logistic growth equation
  10. The non-stiff case of c=2
  11. The stiff case of c=500
  12. The SEIR system
  13. The case Pi=0
  14. The case of nonzero Pi
  15. Conclusions and discussion

Key Result

Theorem 1

If eq:LMM represents a standard finite difference scheme that is consistent and zero-stable, then any corresponding nonstandard finite difference scheme eq:NSLMM (that is, with the same $\alpha_j$ and $\beta_j$ constants) is necessarily consistent. Furthermore, scheme eq:NSLMM is zero-stable provide

Figures (13)

  • Figure 1: The errors of the different methods with different $\varphi$ functions for different values of $\Delta t$for $c=2$.
  • Figure 2: The errors of the different nonstandard methods for different values of $\Delta t$with $c=2$.
  • Figure 3: Methods SSPMS(4,2), NSSPMS(4,2) and NSSPRK(2,2) (upper left), SSPMS(4,3), NSSPMS(4,3) and NSSPRK(3,3) (upper right) and SSPMS(6,4), NSSPMS(6,4) and NSSPRK(10,4) (lower) with timesteps $\Delta t_1=0.5$ and $\Delta t_2=0.02$. For $\Delta t_2$, the schemes are very close to each other, therefore, only one of them is visible.
  • Figure 4: The sufficient bound $\mathcal{B}$ and the "real" bounds for methods NSSPMS(4,2) (upper left), NSSPMS(4,3) (upper right) and NSSPMS(6,4) (lower) for the boundedness property ($\mathcal{B}^*$) and the weak monotonicity ($\widetilde{\mathcal{B}}$) with $c=2$.
  • Figure 5: The errors of the NSSPMS(5,4) method with different $\varphi$ functions for different values of $\Delta t$ for $c=500$.
  • ...and 8 more figures

Theorems & Definitions (23)

  • Theorem 1: anguelov, Theorem 6
  • Lemma 1: Nonstandard form of Taylor's theorem
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 3
  • ...and 13 more