13 research outputs found
Preface to the special issue 'High order methods for CFD problems'
Since a few years, there has been a renew in interest in (very) high order schemes for compressible fluid dynamics for steady and unsteady problems. Within Europe and the US, several conferences deal with these issues, some of them have been launched only recently. Several special issues of recent or future AIAA conferences are specially devoted to that topic. The goal of these researches is to design cheaper and more efficient numerical methods able to handle very large and very complex problems
A high-order nonconservative approach for hyperbolic equations in fluid dynamics
It is well known, thanks to Lax-Wendroff theorem, that the local conservation
of a numerical scheme for a conservative hyperbolic system is a simple and
systematic way to guarantee that, if stable, a scheme will provide a sequence
of solutions that will converge to a weak solution of the continuous problem.
In [1], it is shown that a nonconservative scheme will not provide a good
solution. The question of using, nevertheless, a nonconservative formulation of
the system and getting the correct solution has been a long-standing debate. In
this paper, we show how get a relevant weak solution from a pressure-based
formulation of the Euler equations of fluid mechanics. This is useful when
dealing with nonlinear equations of state because it is easier to compute the
internal energy from the pressure than the opposite. This makes it possible to
get oscillation free solutions, contrarily to classical conservative methods.
An extension to multiphase flows is also discussed, as well as a
multidimensional extension
A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
We are interested in the approximation of a steady hyperbolic problem. In
some cases, the solution can satisfy an additional conservation relation, at
least when it is smooth. This is the case of an entropy. In this paper, we
show, starting from the discretisation of the original PDE, how to construct a
scheme that is consistent with the original PDE and the additional conservation
relation. Since one interesting example is given by the systems endowed by an
entropy, we provide one explicit solution, and show that the accuracy of the
new scheme is at most degraded by one order. In the case of a discontinuous
Galerkin scheme and a Residual distribution scheme, we show how not to degrade
the accuracy. This improves the recent results obtained in [1, 2, 3, 4] in the
sense that no particular constraints are set on quadrature formula and that a
priori maximum accuracy can still be achieved. We study the behavior of the
method on a non linear scalar problem. However, the method is not restricted to
scalar problems
High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics
In the present work, a high order finite element type residual distribution
scheme is designed in the framework of multidimensional compressible Euler
equations of gas dynamics. The strengths of the proposed approximation rely on
the generic spatial discretization of the model equations using a continuous
finite element type approximation technique, while avoiding the solution of a
large linear system with a sparse mass matrix which would come along with any
standard ODE solver in a classical finite element approach to advance the
solution in time. In this work, we propose a new Residual Distribution (RD)
scheme, which provides an arbitrary explicit high order approximation of the
smooth solutions of the Euler equations both in space and time. The design of
the scheme via the coupling of the RD formulation \cite{mario,abg} with a
Deferred Correction (DeC) type method \cite{shu-dec,Minion2}, allows to have
the matrix associated to the update in time, which needs to be inverted, to be
diagonal. The use of Bernstein polynomials as shape functions, guarantees that
this diagonal matrix is invertible and ensures strict positivity of the
resulting diagonal matrix coefficients. This work is the extension of
\cite{enumath,Abgrall2017} to multidimensional systems. We have assessed our
method on several challenging benchmark problems for one- and two-dimensional
Euler equations and the scheme has proven to be robust and to achieve the
theoretically predicted high order of accuracy on smooth solutions
Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes
In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the Residual Distribution method of \cite{ENORD} to very high order of accuracy, by extending the preliminary work discussed in \cite{abgrallLarat} to systems and hybrid meshes. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we an both have a non oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems.Adaptive Schemes for Deterministic and Stochastic Flow Problem