Abstract
We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.
Highlights
As an introduction to this article on structure-preserving integrators for dissipative systems, we first summarize the state-of-the-art of the literature and provide a description of the GENERIC formulation and its properties
It turns out that thermodynamically admissible evolution equations for nonequilibrium systems have a more general and well-defined structure known as GENERIC [17,18,19,20], which possesses the following distinct features: (i) conservation of the total energy; (ii) separation of the reversible and irreversible dynamics; (iii) the reversible dynamics preserves a Poisson structure; (iv) entropy production is unaffected by the reversible dynamics; (v) non-negative entropy production rate
Unlike common approaches that are based on exact energy conservation, we propose in this article a framework to construct structure-preserving integrators for dissipative systems, i.e. GENERIC integrators
Summary
As an introduction to this article on structure-preserving integrators for dissipative systems, we first summarize the state-of-the-art of the literature and provide a description of the GENERIC formulation and its properties. Where x is the set of independent variables required to describe a given nonequilibrium system, E and S represent, respectively, the total energy and entropy as functions of the independent variables x, and L and M denote the antisymmetric Poisson matrix and the positive semidefinite (symmetric) friction matrix, respectively Note that both L and M can depend on the independent variables x so that the fundamental time evolution equation (1.1) could be highly nonlinear. Unlike common approaches that are based on exact energy conservation, we propose in this article a framework to construct structure-preserving integrators for dissipative systems, i.e. GENERIC integrators
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