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Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems

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Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

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Abstract

This chapter focuses on discrete gradient integrators intending to preserve the first integral or the Lyapunov function of the original continuous system. Incorporating the discrete gradients with exponential integrators, we discuss a novel exponential integrator for the conservative or dissipative system \(\dot{y}=Q(My+\nabla U(y))\), where Q is a \(d\times d\) real matrix, M is a \(d\times d\) symmetric real matrix and \(U : \mathbb {R}^{d}\rightarrow \mathbb {R}\) is a differentiable function. For conservative systems, the exponential integrator preserves the energy, while for dissipative systems, the exponential integrator preserves the decaying property of the Lyapunov function. Two properties of the new scheme are presented. Numerical experiments demonstrate the remarkable superiority of the new scheme in comparison with other structure-preserving schemes in the recent literature.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, Jiangsu, China

    Xinyuan Wu

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

    Xinyuan Wu & Bin Wang

Authors
  1. Xinyuan Wu
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  2. Bin Wang
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Correspondence to Xinyuan Wu .

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Wu, X., Wang, B. (2018). Exponential Average-Vector-Field Integrator for Conservative or Dissipative Systems. In: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Singapore. https://doi.org/10.1007/978-981-10-9004-2_2

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