arXiv subjects

François Dubois

Publications and source records attributed to François Dubois.

70 records · Page 4Linked to original sources

arXiv2013

Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves

We follow the mathematical framework proposed by Bouchut and present in this contribution a dual entropy approach for determining equilibrium states of a lattice Boltzmann scheme. This method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law. It appears as a good mathematical framework for establishing a "H-theorem" for the system of equations with discrete velocities. The dual entropy approach is used with D1Q3 lattice Boltzmann schemes for the Burgers equation. It conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition. Satisfactory numerical results for strong nonlinear shocks and rarefactions are presented. We prove also that the dual entropy approach can be applied with a D1Q3 lattice Boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics. We establish also a negative result: with the present framework, the dual entropy approach cannot be used for the shallow water equations.

math.NA↗

arXiv2011

Linear Lattice Boltzmann Schemes for Acoustic: parameters choices and isotropy properties

In this paper, we investigate the numerous parameters choices for linear lattice Boltzmann schemes according to the definition of the isotropic order given in \cite{ADG11}. This property---written in a general framework including all of the \ddqq schemes---can be read through a group operation. It implies some relations on the parameters of the scheme (equilibrium states and relaxation times) that give rigorous methodology to select them according to the desired order of isotropy. For acoustic applications in two spaces dimensions (namely \ddqn and \ddqt schemes) this methodology is used to propose a full description of the sets of parameters that involve isotropy of order $m$ ($m\in\{1,2,3,5\}$ for \ddqn and $m\in\{1,2\}$ for \ddqt). We then propose numerical illustrations for the \ddqn scheme.

math.AP↗

arXiv2011

Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9

In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d'Humiéres. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a lattice Boltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders.

math.NA↗

arXiv2011

Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are explicited in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach.

math.NA↗

arXiv2011

Homographic scheme for Riccati equation

In this paper we present a numerical scheme for the resolution of matrix Riccati equation, usualy used in control problems. The scheme is unconditionnaly stable and the solution is definite positive at each time step of the resolution. We prove the convergence in the scalar case and present several numerical experiments for classical test cases.

math.NA↗

arXiv2011

Lorentz Transform and Staggered Finite Differences for Advective Acoustics

We study acoustic wave propagation in a uniform stationary flow. We develop a method founded on the Lorentz transform and a hypothesis of irrotationality of the acoustic perturbation. After a transformation of the space-time and of the unknown fields, we derive a system of partial differential equations that eliminates the external flow and deals with the classical case of non advective acoustics. A sequel of the analysis is a new set of perfectly matched layers equations in the spirit of the work of Berenger and Collino. The numerical implementation of the previous ideas is presented with the finite differences method HaWAY on cartesian staggered grids. Relevant numerical tests are proposed.

math.NA↗

arXiv2011

Mathematical modeling of antigenicity for HIV dynamics

This contribution is devoted to a new model of HIV multiplication motivated by the patent of one of the authors. We take into account the antigenic diversity through what we define "antigenicity", whether of the virus or of the adapted lymphocytes. We model the interaction of the immune system and the viral strains by two processes. On the one hand, the presence of a given viral quasi-species generates antigenically adapted lymphocytes. On the other hand, the lymphocytes kill only viruses for which they have been designed. We consider also the mutation and multiplication of the virus. An original infection term is derived. So as to compare our system of differential equations with well-known models, we study some of them and compare their predictions to ours in the reduced case of only one antigenicity. In this particular case, our model does not yield any major qualitative difference. We prove mathematically that, in this case, our model is biologically consistent (positive fields) and has a unique continuous solution for long time evolution. In conclusion, this model improves the ability to simulate more advanced phases of the disease.

math.NA↗

arXiv2011

An introduction to finite volumes for gas dynamics

We propose an elementary introduction to the finite volume method in the context of gas dynamics conservation laws. Our approach is founded on the advection equation, the exact integration of the associated Cauchy problem, and the so-called upwind scheme in one space dimension. It is then extended in three directions: hyperbolic linear systems and particularily the system of acoustics, gas dynamics with the help of the Roe matrix and two space dimensions by following the approach proposed by Van Leer. A special emphasis on boundary conditions is proposed all along the text.

math.NA↗

arXiv2011

Unconditionnally stable scheme for Riccati equation

We present a numerical scheme for the resolution of matrix Riccati equation used in control problems. The scheme is unconditionnally stable and the solution is definite positive at each time step of the resolution. We prove the convergence in the scalar case and present several numerical experiments for classical test cases.

math.NA↗

arXiv2011

Partial Riemann problem, boundary conditions, and gas dynamics

We introduce in this contribution the notion of partial Riemann problem. Recall that the Riemann problem describes a shock tube interaction between two given states ; the partial Riemann problem is a generalization of the previous concept and introduces the notion of boundary manifold. In what follows, we first recall very classical notions concerning gas dynamics and the associated Riemann problem. In a second part, we introduce the partial Riemann problem for general systems of conservation laws and proves that this problem admits a solution in some class of appropriate nonlinear waves. In section 3, we recall the linearized analysis with the method of characteristics, introduce the weak formulation of the Dirichlet boundary condition for nonlinear situations in terms of the partial Riemann problem and show that lot of physically relevant situations are described with this theoretical framework. In the last paragraph, we propose a practical implementation of the previous onsiderations with the finite volume method.

math.NA↗

arXiv2011

Conservation Laws Invariant for Galileo Group; Cemracs Preliminary results

We propose a notion of hyperbolic system of conservation laws invariant for the Galileo group of transformations. We show that with natural physical and mathematical hypotheses, such a system conducts to the gas dynamics equations or to exotic systems that are detailed in this contribution to Cemracs 99.

math-ph↗

arXiv2010

Dual Raviart-Thomas mixed finite elements

For an elliptic problem with two space dimensions, we propose to formulate the finite volume method with the help of Petrov-Galerkin mixed finite elementsthat are based on the building of a dual Raviart-Thomas basis.

math.NA↗

arXiv2010

Petrov-Galerkin Finite Volumes

math.NA↗

arXiv2010

Nonlinear Interpolation and Total Variation Diminishing Schemes

The Van Leer approach for the approximation of nonlinear scalar conservation laws is studied in one space dimension. The problem can be reduced to a nonlinear interpolation and we propose a convexity property for the interpolated values. We prove that under general hypotheses the method of lines in well posed in $\ ell^{\infty} \cap {\rm BV} $ and we give precise sufficient conditions to establish that the total variation is diminishing. We observe that the second order accuracy can be maintained even at non sonic extrema. We establish also that both the TVD property and second order accuracy can be maintained after discretization in time with the second order accurate Heun scheme. Numerical illustration for the advection equation is presented.

math.NA↗

arXiv2010

Mathematical model for coupling a quasi-unidimensional perfect flow

Nonlinear acoustics of wind instruments conducts to study unidimensional fluid flows. From physically relevant approximations that are modelized with the thin layer Navier Stokes equations, we propose a coupled model where perfect fluid flow is described by the Euler equations of gas dynamics and viscous and thermal boundary layer is modelized by a linear equation. We describe numerical discretization, validate the associated software by comparison with analytical solutions and consider musical application of strongly nonlinear waves in the trombone.

math.NA↗

arXiv2009

Towards higher order lattice Boltzmann schemes

In this contribution we extend the Taylor expansion method proposed previously by one of us and establish equivalent partial differential equations of DDH lattice Boltzmann scheme at an arbitrary order of accuracy. We derive formally the associated dynamical equations for classical thermal and linear fluid models in one to three space dimensions. We use this approach to adjust relaxation parameters in order to enforce fourth order accuracy for thermal model and diffusive relaxation modes of the Stokes problem. We apply the resulting scheme for numerical computation of associated eigenmodes and compare our results with analytical references.

math.NA↗