Abstract
<abstract> We present a new time discretization scheme adapted to the structure of GENERIC systems. The scheme is based on incremental minimization and is therefore variational in nature. The GENERIC structure of the scheme provides stability and conditional convergence. We show that the scheme can be rigorously implemented in the classical case of the damped harmonic oscillator. Numerical evidence is collected, illustrating the performance of the method and, in particular, the conservation of the energy at the discrete level. </abstract>
Highlights
The aim of this note is to discuss a new variational time-discretization scheme adapted to the structure of General Equations for Non-Equilibrium ReversibleIrreversible Coupling (GENERIC)
Introduced by Grmela & Ottinger, this formulation provides a unified frame for describing the time evolution of physical systems out of equilibrium in presence of reversible and irreversible dynamics [36]
Let y denote the state of a closed, nonequilibrium physical system and let E(y) and S(y) be the corresponding total energy and total entropy, respectively
Summary
The aim of this note is to discuss a new variational time-discretization scheme adapted to the structure of General Equations for Non-Equilibrium ReversibleIrreversible Coupling (GENERIC). The conservation of energy and the quantified dissipative character are the distinguishing traits of the GENERIC system (1.1) To replicate these properties at the discrete level leads to so-called structure-preserving approximations. The discretization of the damped harmonic oscillator is addressed in [37], where the temperature is given by a prescribed heat bath In this case, explicit solutions have to be used in order to specify the GENERIC integrator. The reformulation of dissipative systems in terms of scalar equations as (1.2) is usually referred to as De Giorgi’s Energy-Dissipation Principle [12, 29, 33]. We provide the detailed analysis of a specific case, the damped harmonic oscillator, in which the above-mentioned convergence assumptions can be proved to hold (Section 3). We present numerical experiments assessing the performance of the minimizingmovements scheme and compare it to the classical implicit Euler scheme (Subsection 3.5)
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