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

Design of an essentially non-oscillatory reconstruction procedure in finite-element type meshes

An essentially non oscillatory reconstruction for functions defined on finite element type meshes is designed. Two related problems are studied: the interpolation of possibly unsmooth multivariate functions on arbitary meshes and the reconstruction of a function from its averages in the control volumes surrounding the nodes of the mesh. Concerning the first problem, the behavior of the highest coefficients of two polynomial interpolations of a function that may admit discontinuities of locally regular curves is studied: the Lagrange interpolation and an approximation such that the mean of the polynomial on any control volume is equal to that of the function to be approximated. This enables the best stencil for the approximation to be chosen. The choice of the smallest possible number of stencils is addressed. Concerning the reconstruction problem, two methods were studied: one based on an adaptation of the so called reconstruction via deconvolution method to irregular meshes and one that lies on the approximation on the mean as defined above. The first method is conservative up to a quadrature formula and the second one is exactly conservative. The two methods have the expected order of accuracy, but the second one is much less expensive than the first one. Some numerical examples are given which demonstrate the efficiency of the reconstruction

Some preliminary results on a high order asymptotic preserving computationally explicit kinetic scheme

In this short paper, we intend to describe one way to construct arbitrarily high order kinetic schemes on regular meshes. The method can be arbitrarily high order in space and time, run at least CFL one, is asymptotic preserving and computationally explicit, i.e., the computational costs are of the same order of a fully explicit scheme. We also introduce a non linear stability method that enables to simulate problems with discontinuities, and it does not kill the accuracy for smooth regular solutions

A personal discussion on conservation, and how to formulate it

Since the celebrated theorem of Lax and Wendroff, we know a necessary condition that any numerical scheme for hyperbolic problem should satisfy: it should be written in flux form. A variant can also be formulated for the entropy. Even though some schemes, as for example those using continuous finite element, do not formally cast into this framework, it is a very convenient one. In this paper, we revisit this, introduce a different notion of local conservation which contains the previous one in one space dimension, and explore its consequences. This gives a more flexible framework that allows to get, systematically, entropy stable schemes, entropy dissipative ones, or accomodate more constraints. In particular, we can show that continuous finite element method can be rewritten in the finite volume framework, and all the quantities involved are explicitly computable. We end by presenting the only counter example we are aware of, i.e a scheme that seems not to be rewritten as a finite volume scheme.Comment: Proceedings of FVCA10, https://indico.math.cnrs.fr/event/8972

Evaluating a distance function

Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation of Hamilton Jacobi problems, or the fast sweeping method. Here we make a link with some elliptic problem and propose a very fast way to approximate the distance function

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