Schema explicite des différences finies pour un système d'équations non linéaires du type parabolique avec des conditions aux limites non linéaires
Second-order MUSCL schemes based on Dual Mesh Gradient Reconstruction (DMGR)
We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation...
Sharp stability estimates for hyperbolic conservation laws.
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations , in ; fully coupled by the boundary conditions , on , where is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation . We prove that if blows up in finite time then can fail to blow up if and only if and , which is the same condition as the one for non-simultaneous blow-up in the continuous problem. Moreover,...
Simultaneous vs. non-simultaneous blow-up in numerical approximations of a parabolic system with non-linear boundary conditions
We study the asymptotic behavior of a semi-discrete numerical approximation for a pair of heat equations ut = Δu, vt = Δv in Ω x (0,T); fully coupled by the boundary conditions , on ∂Ω x (0,T), where Ω is a bounded smooth domain in . We focus in the existence or not of non-simultaneous blow-up for a semi-discrete approximation (U,V). We prove that if U blows up in finite time then V can fail to blow up if and only if p11 > 1 and p21 < 2(p11 - 1) , which is the same condition as...
Skipping transition conditions in a posteriori error estimates for finite element discretizations of parabolic equations
In this paper we derive a posteriori error estimates for the heat equation. The time discretization strategy is based on a θ-method and the mesh used for each time-slab is independent of the mesh used for the previous time-slab. The novelty of this paper is an upper bound for the error caused by the coarsening of the mesh used for computing the solution in the previous time-slab. The technique applied for deriving this upper bound is independent of the problem and can be generalized to other time...
Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations.
Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation.
Solutions for the cell cycle in cell lines derived from human tumors.
Solutions to Maxwell equations for an electromagnetic beam injected into a plasma.
Some applications of parabolic differential equations of Allen-Cahn type in image processing
Some new results in multiphase geometrical optics
Some new results in multiphase geometrical optics
In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta...
Some remarks concerning stabilization techniques for convection--diffusion problems
There are many methods and approaches to solving convection--diffusion problems. For those who want to solve such problems the situation is very confusing and it is very difficult to choose the right method. The aim of this short overview is to provide basic guidelines and to mention the common features of different methods. We place particular emphasis on the concept of linear and non-linear stabilization and its implementation within different approaches.
Some tracks in air pollution modelling and simulation.
In this article we discuss some issues related to Air Pollution modelling (as viewed by the authors): subgrid parametrization, multiphase modelling, reduction of high dimensional models and data assimilation. Numerical applications are given with POLAIR, a 3D numerical platform devoted to modelling of atmospheric trace species.
Source terms identification for time fractional diffusion equation.
Stabile Mehrschichtverfahren für parabolische Evolutionsgleichungen.
Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation
We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives the solution in an efficient and unconditionally stable way.
Stability and Convergence of the Difference Schemes for Equations of Isentropic Gas Dynamics in Lagrangian Coordinates
Stability criteria for a class of discrete reaction-diffusion equations.