Symmetries and adjoint-symmetries are two fundamental (coordinate-free) structures of PDE systems. Recent work has developed several new algebraic aspects of adjoint-symmetries: three fundamental actions of symmetries on adjoint-symmetries; a Lie bracket on the set of adjoint-symmetries given by the range of a symmetry action; a generalised Noether (pre-symplectic) operator constructed from any non-variational adjoint-symmetry. These results are illustrated here by considering five examples of physically interesting nonlinear PDE systems – nonlinear reaction-diffusion equations, Navier-Stokes equations for compressible viscous fluid flow, surface-gravity water wave equations, coupled solitary wave equations and a nonlinear acoustic equation.

Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearisation (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra. Solutions of the adjoint linearisation equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether’s theorem, which has a modern generalisation to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-symplectic (Noether) operator, which leads to the construction of symplectic 2-form and Poisson bracket for evolution systems. The generalised KdV equation in potential form is used to illustrate all of the results.

L'expression durative postverbale en chinois mandarin a été traitée dans quatre approches différentes dans la littérature, à savoir l'approche de complément, l'approche de prédicat, la cooccurrence des structures de complément et de prédicat et l'approche d'adjoint. Dans cet article, nous proposons une analyse combinatoire par adjoint et par prédicat en tenant compte des facteurs à la fois syntaxiques et sémantiques non seulement de l'expression durative postverbale, mais aussi des verbes et des objets auxquels les duratifs peuvent s'attacher. Nous avons pour objectif de montrer que ce sont ces facteurs conjoints qui permettent de déterminer la distribution en surface et la position syntaxique de l'expression durative postverbale en chinois mandarin.

An exploratory study is performed to investigate the use of a time-dependent discreteadjoint methodology for design optimization of a high-lift wing configuration augmentedwith an active flow control system. The location and blowing parameters associated with aseries of jet actuation orifices are used as design variables. In addition, a geometricparameterization scheme is developed to provide a compact set of design variablesdescribing the wing shape. The scaling of the implementation is studied using severalthousand processors and it is found that asynchronous file operations can greatly improvethe overall performance of the approach in such massively parallel environments. Threedesign examples are presented which seek to maximize the mean value of the liftcoefficient for the coupled system, and results demonstrate improvements as high as 27%relative to the lift obtained with non-optimized actuation. This lift gain is more thanthree times the incremental lift provided by the non-optimized actuation.

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined thewind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFDhas had its most favorable impact on the aerodynamic design of the high-speed cruiseconfiguration of a transport. This success has raised expectations among aerodynamiciststhat the applicability of CFD can be extended to the full flight envelope. However, thecomplex nature of the flows and geometries involved places substantially increased demandson the solution methodology and resources required. Currently most simulations involveReynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) andDetached Eddy Suimulation (DES) codes are occasionally used for component analysis ortheoretical studies. Despite simplified underlying assumptions, current RANS turbulencemodels have been spectacularly successful for analyzing attached, transonic flows. Whetheror not these same models are applicable to complex flows with smooth surface separation isan open question. A prerequisite for answering this question is absolute confidence thatthe CFD codes employed reliably solve the continuous equations involved. Too often,failure to agree with experiment is mistakenly ascribed to the turbulence model ratherthan inadequate numerics. Grid convergence in three dimensions is rarely achieved. Evenresidual convergence on a given grid is often inadequate. This paper discusses issuesinvolved in residual and especially grid convergence.