Convecting reference frames and invariant numerical models

Abstract

In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al..

The Korteweg–de Vries equation and its symmetry-preserving discretization

“…The projection step is in general accomplished through interpolation and the invariance of the whole solution procedure is guaranteed if the interpolation method used is invariant under the same symmetry group that has been used to construct the numerical scheme itself. This strategy has been successfully adapted for the linear heat equation and the viscous Burgers equation [3,4].…”

“…This can be an important property in practical applications, see e.g. [4] and references therein for applications of this property to hydrodynamics. To numerically verify Galilean invariance in the proposed invariant schemes, we integrate the double soliton solution over a short period of time and apply a boost to the invariant and the non-invariant schemes.…”

“…Quite a few articles devoted to the symmetry adapted discretization of PDEs have appeared over the last 20 years (see e.g. [2][3][4][5][8][9][10][17][18][19]27,28,32,38,40,41,46]). Invariant discretizations of the Korteweg-de Vries (KdV) equation were presented in [12,13,46].…”

Symmetry-Preserving Numerical Schemes

“…For example, new ideas relying on r-adaptivity have been implemented to improve the performance of invariant integrators, [12]. Also, in [10,11] an invariant evolution-projection strategy was introduced and invariant meshless discretization schemes were considered in [6].…”

On the development of symmetry-preserving finite element schemes for ordinary differential equations

“…Most of the work thus far has focused on evolutionary partial differential equations and it is now accepted that in order to preserve symmetries, numerical schemes have to be defined on time-evolving meshes. To avoid mesh tangling and other numerical instabilities, the basic invariant numerical schemes have to be combined with evolution-projection techniques, invariant r-adaptive methods, or invariant meshless discretisations, [2,[4][5][6].…”