A new symmetric linearly implicit exponential integrator preserving polynomial invariants or Lyapunov functions for conservative or dissipative systems

TL;DR

This paper introduces a new symmetric linearly implicit exponential integrator that preserves polynomial invariants or Lyapunov functions, improving long-term simulation accuracy and efficiency for conservative and dissipative systems.

Contribution

The paper proposes a novel integrator that maintains key invariants in stiff equations, offering a computationally efficient alternative to fully implicit methods.

Findings

Preserves polynomial invariants and Lyapunov functions in simulations.

Demonstrates superior speed and stability in long-term numerical experiments.

Effective for both ordinary and partial differential equations.

Abstract

We present a new linearly implicit exponential integrator that preserves the polynomial first integrals or Lyapunov functions for the conservative and dissipative stiff equations, respectively. The method is tested by both oscillated ordinary differential equations and partial differential equations, e.g., an averaged system in wind-induced oscillation, the Fermi-Pasta-Ulam systems, and the polynomial pendulum oscillators. The numerical simulations confirm the conservative properties of the proposed method and demonstrate its good behavior in superior running speed when compared with fully implicit schemes for long-time simulations.