A new high order energy and enstrophy conserving Arakawa-like Jacobian differential operator

Abstract

A new high order Arakawa-like method for the incompressible vorticity equation in two-dimensions has been developed. Mimetic properties such as skewsymmetry, energy and enstrophy conservations for the semi-discretization are proved for periodic problems using arbitrary high order summation-by-parts operators. Numerical simulations corroborate the theoretical findings.

On conservation laws of Navier–Stokes Galerkin discretizations

“…A finite element formulation for energy and helicity conservation was proposed in [28], and in [26] it was discussed how an alternate (but equally valid) definition of helicity could be conserved by skew-symmetric formulations. Similar phenomena happen with other types of discretization methods, and some clever discretizations have been developed which 'bring back' conservation laws lost in standard discretizations, beginning decades ago by Arakawa, Fix, and others, for NSE and related equations [1,2,3,10,12,23,27,32,35]. A common theme for all 'enhanced-physics' based schemes is that the more physics is built into the discretization, the more accurate and stable the discrete solutions are, especially over longer time intervals.In the present work, we aim to develop numerical schemes/formulations that preserve even more conservation laws for the full NSE, beyond just energy.…”

Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations

“…The nonlinear terms must be split into skew-symmetric form for the upcoming discrete analysis. For more details regarding different splitting techniques, see [33,11,47]. Note that the systems (2.3) and (2.5) are symmetric which allows for a straightforward use of the energy method.…”

Supraconservative Finite-Volume Methods for the Euler Equations of Subsonic Compressible Flow

“…Also for the shallow-water equations discrete energy conservation is actively pursued [119,121,136], sometimes in conjunction with one other discrete invariant, e.g. enstrophy [120,122]. Only a few exceptions with more than one discretely conserved invariant, viz.…”