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<p>Two energy quadratization approaches, the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV), provide an effective framework for constructing linearly implicit schemes with guaranteed energy dissipation or conservation. However, extrapolation treatments of nonlinear terms result in non-symmetric discretizations. In this paper, we extend the SAV approach to a symmetric numerical method, aiming to enhance solution quality while preserving computational efficiency, specifically improving numerical stability and computational accuracy. We first develop a general framework of symmetric composition methods preserving quadratic invariants, enabling efficient semi-implicit computation for Hamiltonian systems. Building upon this framework and combining with the SAV approach, we then construct efficient symmetric energy-preserving SAV methods for Hamiltonian PDEs, specifically addressing the nonlinear Schrödinger equation and nonlinear wave equation, with detailed exposition of its high-performance implementation. Finally, we provide several benchmark examples to demonstrate the accuracy, effectiveness and superiority of our numerical methods.</p>

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Highly Efficient Symmetry Energy-Preserving SAV Methods for the Nonlinear Hamiltonian PDEs

  • 期刊论文

  • 发表日期：

    2026年3月27日

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  • Zhuangzhi Xu,
  • Yongzhong Song,
  • Yushun Wang

摘要

Two energy quadratization approaches, the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV), provide an effective framework for constructing linearly implicit schemes with guaranteed energy dissipation or conservation. However, extrapolation treatments of nonlinear terms result in non-symmetric discretizations. In this paper, we extend the SAV approach to a symmetric numerical method, aiming to enhance solution quality while preserving computational efficiency, specifically improving numerical stability and computational accuracy. We first develop a general framework of symmetric composition methods preserving quadratic invariants, enabling efficient semi-implicit computation for Hamiltonian systems. Building upon this framework and combining with the SAV approach, we then construct efficient symmetric energy-preserving SAV methods for Hamiltonian PDEs, specifically addressing the nonlinear Schrödinger equation and nonlinear wave equation, with detailed exposition of its high-performance implementation. Finally, we provide several benchmark examples to demonstrate the accuracy, effectiveness and superiority of our numerical methods.