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We construct a third order multidimensional upwind residual distribution scheme for the system of the Navier–Stokes equations. The underlying approximation is obtained using standard P2 Lagrange finite elements. To discretise the inviscid component of the equations, each element is divided in sub-elements over which we compute a high order residual defined as the integral of the inviscid fluxes on the boundary of the sub-element. The residuals are distributed to the nodes of each sub-element in a multi-dimensional upwind way. To obtain a discretisation of the viscous terms consistent with this multi-dimensional upwind approach, we make use of a Petrov–Galerkin analogy. The analogy allows to find a family of test functions which can be used to obtain a weak approximation of the viscous terms. The performance of this high-order method is tested on flows with high and low Reynolds number.

Residual distribution schemes on curved geometries are discussed in the context of higher order spatial discretization for hyperbolic conservation laws. The discrete solution, defined by a Finite Element space based on triangular Lagrangian Pk elements, is globally continuous. A natural sub-triangulation of these elements allows to reuse the simple distribution schemes previously developed for linear P1 triangles. The paper introduces curved elements with piecewise quadratic and cubic approximation of the boundaries of the domain, using standard sub- or isoparametric transformation. Numerical results for the Euler equations confirm the predicted order of accuracy, showing the importance of a higher order approximation of the geometry.

International audience; We discuss preliminary results on the construction of uniformly high-order residual distribution $(\cal RD)$ type discretizations for steady advection-diffusion on unstructured grids. A properly designed scaling of the $(\cal RD)$ upwind stabilization with the physical viscosity allows to obtain schemes with uniform and arbitrary accuracy, on a very compact stencil. Second- and third-order examples are given to illustrate the potential of the approach.

In this article, the results of aeroacoustic cases obtained by high-order methods are presented. One internal and one external case are considered. The internal case is a transonic cavity. It has been computed by three parters, namely FOI-LiU (Linkoping University), University of Stuttgart (USTUTT) and Dassault Aviation (DASSAV). FOI-LiU has computed with their higher order accurate finite difference solver using a block structured grid. Comparisons are made to reference calculations using an unstructured grid with Edge. USTUTT uses a high order discontinuous Galerkin spectral element code for the final high-order computations and a mixed modal/nodal DG code for the baseline computations. Dassault Aviation uses their in-house stabilized finite element code Aether, both for the reference and the higher-order computations. In both instances unstructured tetrehedral grids are used. The external case is a quasi-two dimensional generic wing and flap configuration defined in the VALIANT FP7 project. Two institutions were involved in the high-order simulation of this low-Mach number test case, M = 0.15, Central Institute of Aero- and Hydrodynamics, TsAGI, and the von Karman Institute, VKI. In the following, the acoustic predictions corresponding to the two test cases will be described.

boundary conditions are applied for subsonic outlet in order to adjust it to the multidimensional upwind discretization of Residual Distribution Method. The article analyzes boundary conditions from the litera- ture and gives suggestions of improvement based on physical considerations. The dierent formulations are compared through analytical problems.

We present an extension of the multidimensional upwind distributive schemes to high order solution spaces. We look into different high-order discretization issues such as: quadratic and cubic boundary curvature; monotonicity of the schemes in presence of solutions with discontinuities; discretisation of temporal terms for unsteady applications and discretization of diffusive fluxes. Results of test cases representative of all these issues are presented.

This article deals with the discretization of Linearized Euler Equations (LEEs) by multidimensional upwind Residual Distribution methods. Linearized Euler equations are applied in the domain where there is no source of sound and the analogy methods such as Ffowcs-Williams can not be used because of gradients in the mean flow. Residual distribution method is a class of schemes that is in between finite-element and finite-volume. In particular, the schemes that we use are multidimensional upwind which make them very attractive because of their very low cross-dissipation. First, we define the formulation of the LEEs that we choose to use. Then, we focus on the residual schemes and we describe two ways of discretizing unsteady problems. The third part presents a wave number of those schemes. Finally, we show the advantage of these schemes on several acoustic problems.

International audience; This article deals with the latest development in the construction of monotone high-order residual distribution schemes (RDS) on quadrilateral meshes. In the first part we give some generalities about the way upwind high-order RDS is designed. In the second part, some standard schemes are described. Then, the design of a stabilization to help the convergence is shown. The last part of this article is dedicated to the results obtained with these schemes.