A Multi-Invariant Preserving Discrete Gradient Methods
This work provides a more robust and accurate numerical integration method for researchers and engineers working with conservative dynamical systems, particularly those requiring the preservation of multiple invariants.
This paper introduces a novel predictor-corrector method that uses discrete gradient corrections to preserve nonlinear invariants in conservative systems. The method maintains the accuracy of the original predictor and extends to problems with multiple invariants, demonstrating improved robustness and conservation in long-term simulations of various dynamical systems.
This work introduces a novel structure-preserving methods for conservative systems based on a predictor-corrector strategy. The framework applies a discrete gradient correction to predictions generated by explicit one-step or multi-step schemes, which preserves nonlinear invariants while maintaining the accuracy order of the original predictor. This approach extends naturally to problems requiring simultaneous conservation of multiple invariants. Under mild conditions, conservation properties, solvability, numerical accuracy, and stability are established. Long-term numerical simulations on Lotka-Volterra systems, sine-Gordon equations, rigid body dynamics, and Kepler problems demonstrate improved robustness and conservation properties compared to existing projection and relaxation methods.