Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

Abstract

<jats:p>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.</jats:p>

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

“…for simplicity), (2) is the kinematic relationship between the strain field ε 3 and v, and the rheological relationship (3) contains, in addition to Young's modulus E, two positive coefficientsÊ, τ. The Poynting-Thomson-Zener model is a subfamily within the Kluitenberg-Verhás model family, which family can be obtained via a nonequilibrium thermodynamical internal variable approach [16].…”

“…the inequality following from (6). For phenomena much slower than these time scales, the rule-of-thumb approximation of keeping only the lowest time derivative for any quantity present in (3) gives the Hooke model 3 In the present context, ε can be used as the thermodynamical state variable for elasticity, but not in general, see [17,18]. 4 Eq.…”

“…Partly inspired by, and partly based on, the intensive development on symplectic schemes for reversible problems, remarkable research is done in recent years on dissipative systems, more on ones with finite degrees of freedom (including [2][3][4][5]) and less for continua (see, e.g., [6][7][8][9]).…”

Wave Propagation in Rocks – Investigating the Effect of Rheology

“…This has led our research group to develop, similarly to recent research activity worldwide [24][25][26][27][28][29][30][31][32][33], some own numerical methods, which outperform the commercially existing ones, in each of these aspects. We intend to benefit from the following principles:…”

“…• the powerful symplectic methods for reversible systems, • the second law of thermodynamics, which acts not only as a consistency requirement but, being realized as the mathematical asymptotic stability property of thermodynamical models, can help in stability of the numerical scheme as well [29][30], and • spacetime aspects encoded in the basic equations of the model: e.g., the balances of mass, of momentum, and of energy are actually a spacetime divergence of a spacetime tensor. Such a finite difference scheme has been successfully built, its resistance against numerical artefacts has been ensured through accuracy and stability analysis, and test runs have proved that the scheme is precise and fast with low memory demand [21], both for Hooke and PTZ models.…”