[Paper Review] An explicit and practically invariants-preserving method for conservative systems

Wenjun Cai, Yuezheng Gong+1 more|arXiv (Cornell University)|Sep 15, 2020

Numerical methods for differential equations43 references4 citations

TL;DR

This paper proposes an explicit invariants-preserving (EIP) method that combines high-order explicit Runge-Kutta schemes with a simplified projection technique to preserve invariants in conservative systems. The method maintains the same order of accuracy as the underlying Runge-Kutta method and achieves invariants error on the order of $\mathcal{O}(h^{2(p+1)})$, reaching machine precision for sufficiently high-order methods.

ABSTRACT

An explicit numerical strategy that practically preserves invariants is derived for conservative systems by combining an explicit high-order Runge-Kutta (RK) scheme with a simple modification of the standard projection approach, which is named the explicit invariants-preserving (EIP) method. The proposed approach is shown to have the same order as the underlying RK method, while the error of invariants is analyzed in the order of $\mathcal{O}\left(h^{2(p+1)} ight),$ where $h$ is the time step and $p$ represents the order of the method. When $p$ is appropriately large, the EIP method is practically invariants-conserving because the error of invariants can reach the machine accuracy. The method is illustrated for the cases of single and multiple invariants, with regard to both ODEs and high-dimensional PDEs. Extensive numerical experiments are presented to verify our theoretical results and demonstrate the superior behaviors of the proposed method in a long time numerical simulation. Numerical results suggest that the fourth-order EIP method preserves much better the qualitative properties of the flow than the standard fourth-order RK method and it is more efficient in practice than the fully implicit integrators.

Motivation & Objective

• To develop an explicit numerical method that preserves invariants in conservative systems without requiring nonlinear solves.
• To overcome the limitation of standard explicit Runge-Kutta methods, which fail to conserve general polynomial invariants beyond linear and quadratic forms.
• To achieve practical invariants conservation by reducing the error in conserved quantities to machine precision through a simplified projection approach.
• To extend the applicability of invariant-preserving methods to high-dimensional PDEs and systems with multiple invariants.
• To demonstrate superior long-time numerical stability and efficiency compared to fully implicit and standard explicit schemes.

Proposed method

• The method combines an explicit high-order Runge-Kutta (RK) scheme with a modified projection technique that avoids solving nonlinear equations.
• The projection step enforces invariants by adjusting the solution using a simplified Lagrange multiplier approach based on the discrete gradient of the invariant.
• The approach is derived as a practical simplification of the standard projection method, preserving the order of accuracy of the underlying RK scheme.
• The method is generalized to handle both single and multiple invariants, including energy and mass in Hamiltonian systems.
• For PDEs, the method is applied after spatial semidiscretization, preserving invariants in high-dimensional systems such as the Gross-Pitaevskii equation.
• The implementation uses a quadratization strategy to reformulate general invariants into quadratic forms, enabling explicit treatment via projection.

Experimental results

Research questions

• RQ1 Can an explicit numerical method preserve general polynomial invariants in conservative systems while maintaining the order of accuracy of the base integrator?
• RQ2 How can the projection-based invariants-preserving approach be simplified to avoid nonlinear solves and remain explicit?
• RQ3 What is the convergence rate of the invariant error in the proposed method, and can it reach machine precision for high-order schemes?
• RQ4 How does the EIP method compare in long-time simulation performance and conservative properties to standard explicit and fully implicit integrators?
• RQ5 Can the method be effectively extended to high-dimensional PDEs with multiple conserved quantities, such as in Bose-Einstein condensates?

Key findings

• The EIP method preserves the same order of accuracy as the underlying explicit Runge-Kutta scheme, with invariant errors on the order of $\mathcal{O}(h^{2(p+1)})$.
• For sufficiently high-order methods (e.g., fourth-order), the invariant error reaches machine precision, making the method practically invariants-conserving.
• Numerical experiments on ODEs and PDEs, including ring and 3D vortex dynamics in Bose-Einstein condensates, confirm long-term conservation of mass and energy to machine precision.
• The fourth-order EIP method outperforms the standard fourth-order RK method in preserving qualitative flow properties over long-time simulations.
• The EIP method is more efficient in practice than fully implicit integrators due to its explicit structure and linear solve requirements.
• The method maintains excellent conservative behavior across multiple invariants and high-dimensional problems, with consistent error trends observed in both 2D and 3D dynamics.