Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems

Abstract

In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system $\dot{y}=Q(My+\nabla U(y))$, where $Q$ is a $d\times d$ skew-symmetric or negative semidefinite real matrix, $M$ is a $d\times d$ symmetric real matrix, and $U:{\mathbb{R}}^d\to \mathbb{R}$ is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.

References in corpus (3)

• Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
• A unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions and/or first integrals
• Locally exact modifications of numerical schemes

Cited by in corpus (15)

• A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation
• On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems
• Functionally-fitted energy-preserving integrators for Poisson systems
• Error analysis of one-stage explicit extended RKN integrators for semilinear wave equations
• Long-term analysis of exponential integrators for highly oscillatory conservative systems
• Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations
• Efficient energy-preserving methods for charged-particle dynamics
• Exponential Discrete Gradient Schemes for Stochastic Differential Equations
• Exponential energy-preserving methods for charged-particle dynamics in a strong and constant magnetic field
• Geometric two-scale integrators for highly oscillatory system: uniform accuracy and near conservations
• A new symmetric linearly implicit exponential integrator preserving polynomial invariants or Lyapunov functions for conservative or dissipative systems
• Optimal convergence and long-time conservation of exponential integration for Schrödinger equations in a normal or highly oscillatory regime
• Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems
• Long-time energy analysis of extended RKN integrators for muti-frequency highly oscillatory Hamiltonian systems
• Arbitrary-order functionally fitted energy-diminishing methods for gradient systems