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Stability and convergence of multi-product expansion splitting methods with negative weights for semilinear parabolic equations

Xianglong Duan, Chaoyu Quan, Jiang Yang, Zijing Zhu

TL;DR

This work addresses the long-standing question of stability for arbitrarily high-order $\mathrm{MPE}$ splitting methods with negative weights applied to nonlinear semilinear parabolic equations. It develops a stability framework by decomposing the splitting into evolution components $\mathcal{E}_A(\tau)$, $\mathcal{E}_B(\tau)$, and $\mathcal{E}_B(\tau)$-$\mathcal{I}$ and employing Lie-derivative calculus, resulting in uniform stability bounds and a convergence theory via consistency in Sobolev spaces. The authors present two construction pathways for high-order MPE schemes (direct order-conditions and Richardson extrapolation), derive two new fourth-order operators, and demonstrate how Richardson extrapolation can yield $2k$-th order schemes from the Strang base. Numerical experiments across toy, Allen–Cahn, Fisher–KPP, Schrödinger, and reaction-diffusion models validate stability, accuracy, adaptivity, and the benefits of MPE over SPE with negative weights, establishing a practical, rigorous foundation for high-order splitting in nonlinear parabolic PDEs.

Abstract

The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolves a long-standing problem -- how to establish the stability of multi-product expansion (MPE) splitting methods with negative weights. The difficulty occurs because negative weights in high-order MPE method cause the sum of the absolute values of weights larger than one, making standard stability proofs fail. In particular, we take the semilinear parabolic equation as a typical model and establish the stability of arbitrarily high-order MPE splitting methods with positive time steps but possibly negative weights. Rigorous convergence analysis is subsequently obtained from the stability result. Extensive numerical experiments validate the stability and accuracy of various high-order MPE splitting methods, highlighting their efficiency and robustness.

Stability and convergence of multi-product expansion splitting methods with negative weights for semilinear parabolic equations

TL;DR

This work addresses the long-standing question of stability for arbitrarily high-order splitting methods with negative weights applied to nonlinear semilinear parabolic equations. It develops a stability framework by decomposing the splitting into evolution components , , and - and employing Lie-derivative calculus, resulting in uniform stability bounds and a convergence theory via consistency in Sobolev spaces. The authors present two construction pathways for high-order MPE schemes (direct order-conditions and Richardson extrapolation), derive two new fourth-order operators, and demonstrate how Richardson extrapolation can yield -th order schemes from the Strang base. Numerical experiments across toy, Allen–Cahn, Fisher–KPP, Schrödinger, and reaction-diffusion models validate stability, accuracy, adaptivity, and the benefits of MPE over SPE with negative weights, establishing a practical, rigorous foundation for high-order splitting in nonlinear parabolic PDEs.

Abstract

The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolves a long-standing problem -- how to establish the stability of multi-product expansion (MPE) splitting methods with negative weights. The difficulty occurs because negative weights in high-order MPE method cause the sum of the absolute values of weights larger than one, making standard stability proofs fail. In particular, we take the semilinear parabolic equation as a typical model and establish the stability of arbitrarily high-order MPE splitting methods with positive time steps but possibly negative weights. Rigorous convergence analysis is subsequently obtained from the stability result. Extensive numerical experiments validate the stability and accuracy of various high-order MPE splitting methods, highlighting their efficiency and robustness.
Paper Structure (21 sections, 13 theorems, 162 equations, 10 figures, 8 tables)

This paper contains 21 sections, 13 theorems, 162 equations, 10 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. MPE splitting methods for general evolutionary problems
  4. Direct construction from order conditions
  5. Construction by Richardson extrapolation
  6. Uniform stability of MPE splitting methods for semilinear parabolic equations
  7. Semilinear parabolic equations
  8. Stability of $\mathcal{E}_A(\tau)$ and $\mathcal{E}_B(\tau)$
  9. Stability of genaral MPE splitting methods
  10. Convergence of MPE splitting methods for semilinear parabolic equations
  11. Stability of $\mathcal{E}(\tau)$
  12. Consistency of MPE splitting methods
  13. Convergence results
  14. Numerical experiments
  15. Accuracy test
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $\mathcal{S}^{[k]}(t)$ be an SPE operator of the form eq:SPE_scheme, which possesses the time reversibility Assume that $\mathcal{S}^{[k]}(t)$ is expanded as where the error terms $E_{i}$ are the nested commutators of $D_A$ and $D_B$ depending on the specific form of $\mathcal{S}^{[k]}(t)$. Then, the following holds:

Figures (10)

  • Figure 1: Numerical solution to the AC equation \ref{['eq:AC']} by the $\mathcal{S}^{[2],1}$ (left), $\mathcal{S}^{[4],4}$ (middle), and $\mathcal{S}^{[4]}_{\rm neg}$ (right) splitting operators with time step $\tau = 1/40$ at different time.
  • Figure 2: The temporal evolution of energy (left) and evolution of maximum norm (right) of the three operator splitting methods with $\tau = 1/40$ until $T=10$.
  • Figure 3: Solution snapshots of the conservative Allen--Cahn equation using the adaptive time strategy of the $\mathcal{S}^{[4],3}$ form at different time.
  • Figure 4: Evolutions of the original energy (a), maximum of numerical solution (b), mass (c) and time step size (d) of the conservative Allen--Cahn equation using different operator splitting adaptive time strategies compared with uniform time step strategy until time $T=60$.
  • Figure 5: Numerical solutions to the Fisher-KPP equation by the $\mathcal{S}^{[4],1}$ splitting operator with splitting step $\tau = 0.001$ at different time.
  • ...and 5 more figures

Theorems & Definitions (26)

  • Lemma 3.1: Time reversibility yoshida1990construction
  • Lemma 4.1: Stability of $\mathcal{E}_A(\tau)$
  • proof
  • Lemma 4.2: Stability of $\mathcal{E}_B(\tau)-\mathcal{I}$
  • proof
  • Lemma 4.3: Stability of $\mathcal{E}_B(\tau)$
  • proof
  • Lemma 4.4: Stability of $\mathcal{E}_B(\tau)$
  • proof
  • Lemma 4.5: Stability of $\mathcal{E}_B(\tau)-\mathcal{I}$
  • ...and 16 more