Time evolution equations for non-equilibrium systems have a well-defined mathematical structure in which reversible and irreversible contributions are identified separately. In particular, the reversible contribution is generally assumed to be of the Hamiltonian form and hence comes with an underlying geometric structure. This requirement imposes severe restrictions on the admissible form of the time evolution equations which express the idea that the reversible time evolution should be “under mechanistic control.” Additional features of thermodynamic systems are the conservation of entropy under reversible dynamics and the non-negativity of entropy production under irreversible dynamics. The present paper addresses the problem of how the rich structure of thermodynamically admissible evolution equations can be preserved under time discretization, which is the key to successful numerical calculations.

Our discussion is based on the GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) formulation of time evolution equations for non-equilibrium systems [2], [3], [4],
$$\begin{array}{c}{\displaystyle \frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{t}}}=\mathit{L}{\displaystyle \frac{\partial \mathit{E}}{\partial \mathit{x}}}+\mathit{M}{\displaystyle \frac{\partial \mathit{S}}{\partial \mathit{x}}}\mathrm{,}\end{array}$$(1)
where *x* represents the set of independent variables required for an autonomous description of a given non-equilibrium system, *E* and *S* are the total energy and entropy expressed in terms of the variables *x*, and *L* and *M* are certain linear operators, or matrices, which can depend on *x*. We here use the notation for finite-dimensional systems (for example, partial rather than functional derivatives) because it is advantageous to develop the basic ideas for GENERIC time integration for systems with only a few degrees of freedom. The two contributions to the time evolution of *x* generated by the energy *E* and the entropy *S* in eq. (1) are called the reversible and irreversible contributions, respectively. Equation (1) is supplemented with the complementary degeneracy requirements
$$\begin{array}{c}\mathit{L}{\displaystyle \frac{\partial \mathit{S}}{\partial \mathit{x}}}=0\end{array}$$(2)
and
$$\begin{array}{c}\mathit{M}{\displaystyle \frac{\partial \mathit{E}}{\partial \mathit{x}}}=0.\end{array}$$(3)
The requirement that the entropy gradient $\partial \mathit{S}\mathrm{/}\partial \mathit{x}$ is in the null-space of *L* in eq. (2) expresses the reversible nature of the Hamiltonian contribution to the dynamics: the functional form of the entropy is such that it cannot be affected by the operator generating reversible dynamics. The requirement that the energy gradient $\partial \mathit{E}\mathrm{/}\partial \mathit{x}$ is in the null-space of *M* in eq. (3) expresses the conservation of the total energy in a closed system by the irreversible contribution to dynamics.

Further general properties of *L* and *M* are discussed most conveniently in terms of the two brackets $$\begin{array}{rl}\{\mathit{A}\mathrm{,}\mathit{B}\}& ={\displaystyle \frac{\partial \mathit{A}}{\partial \mathit{x}}}\xb7\mathit{L}{\displaystyle \frac{\partial \mathit{B}}{\partial \mathit{x}}}\mathrm{,}\end{array}$$(4)$$\begin{array}{rl}[\mathit{A}\mathrm{,}\mathit{B}]& ={\displaystyle \frac{\partial \mathit{A}}{\partial \mathit{x}}}\xb7\mathit{M}{\displaystyle \frac{\partial \mathit{B}}{\partial \mathit{x}}}\mathrm{,}\end{array}$$(5) where *A*, *B* are sufficiently regular, real-valued functions defined on the space of independent variables and the dot indicates a canonical product. In terms of these brackets, eq. (1) leads to the following time evolution equation of an arbitrary function *A* generated by *E* and *S*:
$$\begin{array}{c}{\displaystyle \frac{\mathit{d}\mathit{A}}{\mathit{d}\mathit{t}}}=\{\mathit{A}\mathrm{,}\mathit{E}\}+[\mathit{A}\mathrm{,}\mathit{S}].\end{array}$$(6)
Further conditions for *L* can now be stated as the antisymmetry property
$$\begin{array}{c}\{\mathit{A}\mathrm{,}\mathit{B}\}=-\{\mathit{B}\mathrm{,}\mathit{A}\}\mathrm{,}\end{array}$$(7)
the product or Leibniz rule
$$\begin{array}{c}\{\mathit{A}\mathit{B}\mathrm{,}\mathit{C}\}=\mathit{A}\{\mathit{B}\mathrm{,}\mathit{C}\}+\mathit{B}\{\mathit{A}\mathrm{,}\mathit{C}\}\mathrm{,}\end{array}$$(8)
and the Jacobi identity
$$\begin{array}{c}\{\mathit{A}\mathrm{,}\{\mathit{B}\mathrm{,}\mathit{C}\}\}+\{\mathit{B}\mathrm{,}\{\mathit{C}\mathrm{,}\mathit{A}\}\}+\{\mathit{C}\mathrm{,}\{\mathit{A}\mathrm{,}\mathit{B}\}\}=0\mathrm{,}\end{array}$$(9)
where *C* is another sufficiently regular, real-valued function defined on the state space. These properties are well known from the Poisson brackets of classical mechanics and they express the essence of reversible dynamics.

Further properties of *M* can be formulated in terms of the symmetry condition
$$\begin{array}{c}[\mathit{A}\mathrm{,}\mathit{B}]=[\mathit{B}\mathrm{,}\mathit{A}]\end{array}$$(10)
and the non-negativity condition
$$\begin{array}{c}[\mathit{A}\mathrm{,}\mathit{A}]\ge 0.\end{array}$$(11)
The symmetry condition is a generalization of the Onsager symmetry of linear irreversible thermodynamics [5], [6], [7] to non-linear problems (we here do not consider the possibility of Casimir symmetry [5], [8]). The non-negativity condition, together with the degeneracy requirement (2), guarantees that the entropy is a non-decreasing function of time and that entropy production results only from irreversible processes,
$$\begin{array}{c}{\displaystyle \frac{\mathit{d}\mathit{S}}{\mathit{d}\mathit{t}}}={\displaystyle \frac{\partial \mathit{S}}{\partial \mathit{x}}}\xb7\mathit{M}{\displaystyle \frac{\partial \mathit{S}}{\partial \mathit{x}}}=[\mathit{S}\mathrm{,}\mathit{S}]\ge 0.\end{array}$$(12)
The properties (10) and (11) correspond to the symmetry and the positive-semidefiniteness of *M*. From a physical point of view, *M* may be regarded as a friction matrix.

The Jacobi identity (9), which is a highly restrictive condition for formulating proper reversible dynamics, expresses the time–structure invariance of Poisson brackets. As it is tedious to verify the Jacobi identity, symbolic software has been developed for that purpose [9], [10]. It is natural to consider integrators which leave the underlying Poisson bracket invariant even under finite time steps. This idea is well known from the symplectic integrators used in classical mechanics [11]. Symplectic integrators are known for better performance in numerical calculations and hence provide the motivation for introducing GENERIC integrators to perform better calculations for non-equilibrium systems in an analogous way. Even numerical methods that focus only on the proper treatment of entropy can have significant advantages, as has been illustrated for the Lattice Boltzmann method [12], [13] and for the discussion of shock waves in terms of thirteen-moment equations [14].

The question of structure preserving integrators for dissipative systems has been addressed by several groups working on numerical mathematics and is a topic of ongoing work (see, *e. g.*, [15], [16], [17] and references therein). In the mathematical literature, the GENERIC structure is also known as metriplectic structure, and metriplectic integrators have been mentioned in Section V of [15]. In general, however, the characterization of structure preserving integrators seems to be different from the one given in the present paper. In particular, the metriplectic integrators proposed in [18] do not qualify as GENERIC integrators. Whereas the spatial discretization is done in a fully structure preserving manner (in the same spirit as the discretization of hydrodynamics in [19]), the temporal discretization aims at strict discrete conservation laws for energy (and momentum) and at proper monotonicity of the entropy.

We first review the ideas of symplectic time integration and develop the requirements to be imposed on GENERIC integrators. We then construct GENERIC integrators for systems with a single dissipative process. After discussing some details for the specific example of a harmonic oscillator with friction, we summarize the general features of GENERIC integrators and identify some possible directions for future work.

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