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A New Computational Foundation of Thermodynamics

We present a deterministic foundation of thermodynamics for slightly viscous fluids or gases based on a 1st Law in the form of the Euler equations expressing conservation of mass, momentum and energy, and a 2nd Law formulated in terms of kinetic energy, internal (heat) energy, work and shock/turbulent dissipation, without reference to entropy. The Euler equations are regularized in computational solution by a least-squares stabilized finite element method referred to as EG2. The 2nd Law expresses an irreversible transfer of kinetic energy to heat energy in shock/turbulent dissipation arising because the Euler equations lack pointwise solutions. The 2nd Law explains the occurence of irreversibility in formally reversible systems as an effect of instability with blow-up of Euler residuals combined with finite precision computation, without resort to statistical mechanics or ad hoc viscous regularization. EG2 includes a duality-based posteriori error control showing that mean-value outputs are computable to tolerances of interest (while point values are not).

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