Conservation laws in discrete geometry

Margolin Len G.; Baty Roy S.

doi:10.3934/jgm.2019010

Article Contents

2019, Volume 11, Issue 2: 187-203. Doi: 10.3934/jgm.2019010

This issue Previous Article Riemann-Hilbert problem, integrability and reductions Next Article Embedding Camassa-Holm equations in incompressible Euler

Conservation laws in discrete geometry

Margolin Len G.^1, , and
Baty Roy S.^2,

1.
Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
2.
Theoretical Design Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

^* Corresponding author: L. G. Margolin
^* Corresponding author: L. G. Margolin

Received: March 31, 2018

Revised: December 31, 2018

Published: May 2019

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Abstract

The small length scales of the dissipative processes of physical viscosity and heat conduction are typically not resolved in the numerical simulation of high Reynolds number flows in the discrete geometry of computational grids. Historically, the simulations of flows with shocks and/or turbulence have relied on solving the Euler equations with dissipative regularization. In this paper, we begin by reviewing the regularization strategies used in shock wave calculations in both a Lagrangian and an Eulerian framework. We exhibit the essential similarities with Large Eddy Simulation models of turbulence, namely that almost all of these depend on the square of the size of the computational cell. In our principal result, we justify that dependence by deriving the evolution equations for a finite-sized volume of fluid. Those evolution equations, termed finite scale Navier-Stokes (FSNS), contain dissipative terms similar to the artificial viscosity first proposed by von Neumann and Richtmyer. We describe the properties of FSNS, provide a physical interpretation of the dissipative terms and show the connection to recent concepts in fluid dynamics, including inviscid dissipation and bi-velocity hydrodynamics.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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