Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems

TL;DR

The paper develops two core classes of structure-preserving time discretisations. First, it constructs arbitrary-order conservative integrators for general ODEs that exactly preserve all invariants by introducing auxiliary gradient projections and an alternating-form representation of the vector field. Second, it extends to GENERIC systems to produce energy-conserving and entropy-dissipating schemes for both ODEs and PDEs, employing consistent extensions of the Poisson and friction operators with auxiliary variables. Through detailed numerical experiments on the Kepler problem, Kovalevskaya top, internal-combustion engine models, Boltzmann dynamics, and the BBM equation, the authors demonstrate improved fidelity of invariant preservation and thermodynamic consistency over conventional methods. The work leverages the Andrews–Farrell framework and provides practical pathways for high-fidelity long-time simulations of conservative and GENERIC systems, with openly available code for reproducibility.

Abstract

Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and dissipation laws on discretisation in time can yield vastly better approximations for the same computational effort, compared to schemes that are not structure-preserving. In this work we present two novel contributions: (i) an arbitrary-order time discretisation for general conservative ordinary differential equations that conserves all known invariants and (ii) an energy-conserving and entropy-dissipating scheme for both ordinary and partial differential equations written in the GENERIC format, a superset of Poisson and gradient-descent systems. In both cases the underlying strategy is the same: the systematic introduction of auxiliary variables, allowing for the replication at the discrete level of the proofs of conservation or dissipation. We illustrate the advantages of our approximations with numerical examples of the Kepler and Kovalevskaya problems, a combustion engine model, and the Benjamin-Bona-Mahony equation.

Figures (11)

• Figure 1: Trajectories of the Kepler problem. The exact solution is given by the dashed ellipse. All schemes are of the same order.
• Figure 2: Error in scalar invariants of the Kepler problem: $H$, $L$ and $\theta$.
• Figure 3: Error in the position of the orbital body at $t = 2\pi$ for varying timesteps $\Delta t \in 2\pi \cdot 2^k$, $k \in \{-5, \dots, -12\}$ and stages $s \in \{1, \dots, 4\}$. The convergence curve for $s \in \{3, 4\}$ flattens out at smaller timesteps due to round-off error and solver tolerances. Triangles demonstrate observed convergence rates of $2s$.
• Figure 4: Trajectories in $\mathbf{n}, \mathbf{l}$ of the Kovalevskaya top, with implicit midpoint (left) and our proposed scheme (right).
• Figure 5: Error $K - K(0)$ within the implicit midpoint simulation of the Kovalevskaya top.

Theorems & Definitions (16)

• Proposition 2.1: Conservation properties of \ref{['eq:odecpgmod']}
• proof
• Remark 2.2
• Theorem 2.3: Identification of alternating forms
• proof
• Theorem 2.4: Conservation properties of \ref{['eq:integrable_avcpg']}
• proof
• Remark 2.5
• Remark 3.2
• Theorem 3.3: Energy and entropy stability of \ref{['eq:generic_ode_avcpg']}