Energy-conserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions

Umeda, Akihiro; Wakasugi, Yuta; Yoshikawa, Shuji

doi:10.1016/j.apnum.2021.08.009

Cited by 3 publications

(1 citation statement)

References 15 publications

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“…For its vital role of an energy-conserving technique in different systems, many applications and energy-conserving methods have been presented, e.g., a 3D stochastic nonlinear Schrödinger equation with multiplicative noise [47], a nonlinear Schrödinger equation [48], nonlinear wave equations with dynamic boundary conditions [49], nonlinear space fractional Schrödinger equations with a wave operator [50], the pitch-angle scattering in magnetized plasmas [51], the nonlinear Dirac equation [52], the linear wave equation [53], the Rosenau-type equation [54], nonlinear fourth-order wave equations [55], the multidimensional Hermite-DG discretization of Vlasov-Maxwell equations [56], the Vlasov-Ampère system [57], the Vlasov-Ampère system with an exact curl-free constraint [58], relativistic Vlasov-Maxwell equations of laser-plasma interaction [59], the linear wave equation with forcing terms [60], Hamiltonian systems (including the high amplitude vibration of strings and plates) [61], the multi-dimensional Vlasov-Maxwell system [62], generalized nonlinear fractional Schrödinger wave equations [63].…”

Section: Introductionmentioning

confidence: 99%

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

Liu,

Kuo,

Chang

2024

Vibration

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In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of O(h4) for a small time stepsize h.

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