A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

TL;DR

This paper presents a mimetic dual-field discretization method for 3D incompressible Navier-Stokes equations that conserves mass, energy, and helicity, with proven properties and supporting numerical tests.

Contribution

It introduces a novel dual-field mixed weak formulation with a staggered temporal scheme that ensures conservation properties at the discrete level.

Findings

Conservation of mass, energy, and helicity proven for the discretization.

The method produces linear, decoupled algebraic systems.

Numerical tests validate the theoretical properties.

Abstract

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case is also proven. Numerical tests supporting the method are provided.