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The weighted and shifted seven-step BDF method for parabolic equations

Georgios Akrivis, Minghua Chen, Fan Yu

TL;DR

This work resolves the instability of the seven-step backward difference method for parabolic problems by constructing a stable linear combination with its shifted version, forming the weighted and shifted seven-step BDF (WSBDF7) method. Using an expanded energy technique and carefully designed multipliers, the authors prove $A(\varphi)$-stability for parabolic equations and derive robust stability bounds, alongside a detailed multiplier-determination framework. They also provide seventh-order accuracy results under suitable starting-value assumptions and validate the theory with numerical experiments on a heat equation, demonstrating strong stability and high-order convergence for suitable weights $\vartheta$. The findings extend to mean curvature flow, gradient flows, fractional and nonlinear equations, offering a high-order, stable time-stepping tool for a broad class of parabolic problems.

Abstract

Stability of the BDF methods of order up to five for parabolic equations can be established by the energy technique via Nevanlinna--Odeh multipliers. The nonexistence of Nevanlinna--Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in [Akrivis et al., SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472] and covers all six stable BDF methods. The seven-step BDF method is unstable for parabolic equations, since it is not even zero-stable. In this work, we construct and analyze a stable linear combination of two non zero-stable schemes, the seven-step BDF method and its shifted counterpart, referred to as WSBDF7 method. The stability regions of the WSBDF$q, q\leqslant 7$, with a weight $\vartheta\geqslant1$, increase as $\vartheta$ increases, are larger than the stability regions of the classical BDF$q,$ corresponding to $\vartheta=1$. We determine novel and suitable multipliers for the WSBDF7 method and establish stability for parabolic equations by the energy technique. The proposed approach is applicable for mean curvature flow, gradient flows, fractional equations and nonlinear equations.

The weighted and shifted seven-step BDF method for parabolic equations

TL;DR

This work resolves the instability of the seven-step backward difference method for parabolic problems by constructing a stable linear combination with its shifted version, forming the weighted and shifted seven-step BDF (WSBDF7) method. Using an expanded energy technique and carefully designed multipliers, the authors prove -stability for parabolic equations and derive robust stability bounds, alongside a detailed multiplier-determination framework. They also provide seventh-order accuracy results under suitable starting-value assumptions and validate the theory with numerical experiments on a heat equation, demonstrating strong stability and high-order convergence for suitable weights . The findings extend to mean curvature flow, gradient flows, fractional and nonlinear equations, offering a high-order, stable time-stepping tool for a broad class of parabolic problems.

Abstract

Stability of the BDF methods of order up to five for parabolic equations can be established by the energy technique via Nevanlinna--Odeh multipliers. The nonexistence of Nevanlinna--Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in [Akrivis et al., SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472] and covers all six stable BDF methods. The seven-step BDF method is unstable for parabolic equations, since it is not even zero-stable. In this work, we construct and analyze a stable linear combination of two non zero-stable schemes, the seven-step BDF method and its shifted counterpart, referred to as WSBDF7 method. The stability regions of the WSBDF, with a weight , increase as increases, are larger than the stability regions of the classical BDF corresponding to . We determine novel and suitable multipliers for the WSBDF7 method and establish stability for parabolic equations by the energy technique. The proposed approach is applicable for mean curvature flow, gradient flows, fractional equations and nonlinear equations.
Paper Structure (11 sections, 9 theorems, 145 equations, 3 figures, 1 table)

This paper contains 11 sections, 9 theorems, 145 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Stability regions
  3. Multipliers for multistep methods
  4. Multipliers for the WSBDF7 method \ref{['ab-3']}
  5. On the determination of multipliers
  6. Stability
  7. Proof of Theorem \ref{['Th:stab']}
  8. Proof of Theorem \ref{['Th:stab2']}
  9. Proof of Corollary \ref{['Co:stab']}
  10. Error estimates
  11. Numerical results

Key Result

Theorem 1.1

Let $u^0, u^1, \dotsc, u^6\in V.$ The WSBDF7 method ab-3 is stable in the sense that Here $C$ denotes a generic constant, independent of $T$ and the operator $A$ as well as of $f, \tau,$ and $n$.

Figures (3)

  • Figure 2.1: The stability regions $($upper panel$)$ as well as zoom in around the origin $($lower panel$)$, in light blue, for $\vartheta=1,3,10$, respectively.
  • Figure 4.1: The graph of polynomial $P$ of \ref{['Real6']} in the interval $[0,1]$.
  • Figure 4.2: The graph of the polynomial $g-\tilde{c}_\star$ of \ref{['1.22']} in the interval $[-0.4,1]$, left, and zoom in in the interval $[-0.17,0],$ right.

Theorems & Definitions (20)

  • Theorem 1.1: Stability of method \ref{['ab-3']}
  • Theorem 1.2: Second stability estimate
  • Corollary 1.1: Third stability estimate
  • Remark 2.1: The limit of the stability angles $\tilde{\varphi}_q$ of WSBDF$q$
  • Lemma 3.1: D; see also BC and HW
  • Definition 3.1: Multipliers
  • Remark 3.1: Equivalent version of the conditions on the pair $(\beta,\mu)$
  • Lemma 3.2: Equivalent versions of conditions \ref{['A-alpha-mu']} and \ref{['A-beta-mu']}
  • proof
  • Remark 3.2: BDF methods
  • ...and 10 more