34 research outputs found
A semi-lagrangian scheme for hamilton-jacobi-bellman equations on networks
No full textWe present a semi-Lagrangian scheme for the approximation of a class of Hamilton- Jacobi-Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. We prove a convergence result and two error estimates. For an HJB equation with space-independent Hamiltonian, we obtain a first order error estimate. In the general case, we provide, under a hyperbolic CFL condition, a convergence estimate of order one half. The theoretical results are discussed and validated in a numerical tests section
A Semi-Lagrangian Scheme for Hamilton--Jacobi--Bellman Equations on Networks
No full textWe present a semi-Lagrangian scheme for the approximation of a class of Hamilton--Jacobi--Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. We prove a convergence result and two error estimates. For an HJB equation with space-independent Hamiltonian, we obtain a first order error estimate. In the general case, we provide, under a hyperbolic CFL condition, a convergence estimate of order one half. The theoretical results are discussed and validated in a numerical tests section
A non-monotone Fast Marching scheme for a Hamilton-Jacobi equation modelling dislocation dynamics
No full textIn this paper we introduce an extension of the Fast Marching Method introduced by Sethian [6] for the eikonal equation modelling front evolutions in normal direction. The new scheme can deal with a time-dependent velocity without any restriction on its sign. This scheme is then used for solving dislocation dynamics problems in which the velocity of the front depends on the position of the front itself and its sign is not restricted to be positive or negative
Convergence of a generalized fast-marching method for an eikonal equation with a velocity-changing signn
No full textWe present a new fast-marching algorithm for an eikonal equation with a velocity changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time- dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in R^2
Dislocation Dynamics: a Non-local Moving Boundary
No full textIn this article, we present briefly a mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation. We present some partial existence and uniqueness results. Finally, we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion.
An error estimate for a new scheme for mean curvature motion
No full textInternational audienceIn this work, we propose a new numerical scheme for the anisotropic mean curvature equation. The solution of the scheme is not unique, but for all numerical solutions, we provide an error estimate between the continuous solution and the numerical approximation. This error estimate is not optimal, but as far as we know, this is the first one for mean curvature type equation. Our scheme is also applicable to compute the solution to dislocations dynamics equation
L'humanisme juridique à Toulouse au milieu du XVIe siècle : Mesnard (P.). - Jean Bodin à Toulouse, dans Bibliothèque d'Humanisme et Renaissance, t. ХII, 1950, pp. 31-60. - Un rival heureux de Jean Bodin et de Cujas : Etienne Forcadel, dans Zeitschrift der Savigny-Stiftung für Rechtsgeschichte. - Romanische Abtilung, t. LXVII, 1950, pp. 440-458
No full textRenouard Yves. L'humanisme juridique à Toulouse au milieu du XVIe siècle : Mesnard (P.). - Jean Bodin à Toulouse, dans Bibliothèque d'Humanisme et Renaissance, t. ХII, 1950, pp. 31-60. - Un rival heureux de Jean Bodin et de Cujas : Etienne Forcadel, dans Zeitschrift der Savigny-Stiftung für Rechtsgeschichte. - Romanische Abtilung, t. LXVII, 1950, pp. 440-458. In: Annales du Midi : revue archéologique, historique et philologique de la France méridionale, Tome 62, N°12, 1950. pp. 357-358
Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics.
No full textInternational audienceIn this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two
A Generalized Fast Marching Method for dislocation dynamics
No full textInternational audienceIn this paper, we consider a Generalized Fast Marching Method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hyper-surface in (with for physical applications) is given by its normal velocity which is a non-local function of the whole shape of the hyper-surface itself. For this dynamics, we show a convergence result of the GFMM as the mesh size goes to zero. We also provide some numerical simulations in dimension
Beyond uniqueness: Relaxation calculus of junction conditions for coercive Hamilton-Jacobi equations
No full textInternational audienceA junction is a particular network given by the collection of half lines glued together at the origin. On such a junction, we consider evolutive Hamilton-Jacobi equations with coercive Hamiltonians. Furthermore,we consider a general desired junction condition at the origin, given by some monotone function .There is existence and uniqueness of solutions which only satisfy weakly the junction condition (at the origin, they satisfy either the desired junction condition or the PDE).We show that those solutions satisfy strongly a relaxed junction condition (that we can recognize as an effective junction condition). It is remarkable that this relaxed condition can be computed in three different but equivalent ways: 1) using viscosity inequalities, 2) using Godunov fluxes, 3) using Riemann problems.Our result goes beyond uniqueness theory, in the following sense: solutions to two different desired junction conditions and do coincide if
