In this paper we study a discrete Raman laser amplification model given as a Lotka-Volterra system. We show that in an ideal situation, the equations can be written as a Poisson system with boundary conditions using a global change of coordinates. We address the questions of existence and uniqueness of a solution. We deduce numerical schemes for the approximation of the solution that have good stability.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

In this paper, we present some interesting connections between anumber of Riemann-solver free approaches to the numerical solutionof multi-dimensional systems of conservation laws. As a main part,we present a new and elementary derivation of Fey's Method ofTransport (MoT) (respectively the second author's ICE version ofthe scheme) and the state decompositions which form the basis of it.The only tools that we use are quadrature rules applied to themoment integral used in the gas kinetic derivation of the Eulerequations from the Boltzmann equation, to the integration in timealong characteristics and to space integrals occurring in the finitevolume formulation. Thus, we establish a connection between theMoT approach and the kinetic approach. Furthermore,Ostkamp's equivalence result between her evolution Galerkin schemeand the method of transport is lifted up from the level ofdiscretizations to the level of exact evolution operators,introducing a new connection between the MoT and theevolution Galerkin approach. At the same time, we clarifysome important differences between these two approaches.

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

We investigate different asymptotic regimesfor Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.

Intracellular transport in eukarya is attributed to motor proteins that transduce chemical energy into directed mechanical energy. This suggests that, in nonequilibrium systems, fluctuations may be oriented or organized to do work. Here we seek to understand how this is manifested by quantitative mathematical portrayals of these systems.

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 indimension dd ≥ 1. We use a wavelet based sparse gridspace discretization with mesh-width h and order pd ≥ 1, andhp discontinuous Galerkin time-discretization of order $r =O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many timesteps. The linear systems in each time step are solved iterativelyby $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2(Ω)-error of O(N-p) for u(x,T) where N is the total number of operations,provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ with ε > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm thetheory.

A linearly convergent iterative algorithm that approximates the rank-1 convex envelope $f^{rc}$ of a given function $f:\mathbb{R}^{n\times m} \to \mathbb{R}$, i.e. the largest function below f which is convex along all rank-1 lines, is established. The proposed algorithm is a modified version of an approximation scheme due to Dolzmann and Walkington.

This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids23 (1994) 1049–1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN35 (2001) 107–127].

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictatesthe interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.

In this paper we consider a hyperbolic-parabolic problem that models the longitudinal deformations of a thermoviscoelastic rod supported unilaterally by an elastic obstacle. The existence and uniqueness of a strong solution is shown. A finite element approximation is proposed and its convergence is proved. Numerical experiments are reported.

We deal with numerical analysis and simulations of the Davey-Stewartson equations which model, for example, the evolution of water surface waves. This time dependent PDE system is particularly interesting as a generalization of the 1-d integrable NLS to 2 space dimensions. We use a time splitting spectral method where we give a convergence analysis for the semi-discrete version of the scheme. Numerical results are presented for various blow-up phenomena of the equation, including blowup of defocusing, elliptic-elliptic Davey-Stewartson systems and simultaneous blowup at multiple locations in the focusing elliptic-elliptic system. Also the modeling of exact soliton type solutions for the hyperbolic-elliptic (DS2) system is studied.

This article presents the derivation of a semi-classical model of electromagnetic-wave propagation in a non centro-symmetric crystal. It consists of Maxwell's equations for the wave field coupled with a version of Bloch's equations which takes fully into account the discrete symmetry group of the crystal. The model is specialized in the case of a KDP crystal for which information about the dipolar moments at the Bloch level can be recovered from the macroscopic dispersion properties of the material.

The Method of Fundamental Solutions (MFS) is a boundary-typemeshless method for the solution of certain elliptic boundaryvalue problems. In this work, we investigate the properties of thematrices that arise when the MFS is applied to theDirichlet problem for Laplace's equation in a disk. In particular,we study the behaviour of the eigenvalues of these matrices andthe cases in which they vanish. Based on this, we propose amodified efficient numerical algorithm for the solution of theproblem which is applicable even in the cases when the MFS matrixmight be singular. We prove the convergence of the method foranalytic boundary data and perform a stability analysis of the methodwith respect to the distance of the singularities from the originand the number of degrees of freedom. Finally, wetest the algorithm numerically.

We analyze an isoparametric finite element method to compute the vibration modes of a plate, modeled by Reissner-Mindlin equations, in contact with a compressible fluid, described in terms of displacement variables. To avoid locking in the plate, we consider a low-order method of the so called MITC (Mixed Interpolation of Tensorial Component) family on quadrilateral meshes. To avoid spurious modes in the fluid, we use a low-order hexahedral Raviart-Thomas elements and a non conforming coupling is used on the fluid-structure interface. Applying a general approximation theory for spectral problems, under mild assumptions, we obtain optimal order error estimates for the computed eigenfunctions, as well as a double order for the eigenvalues. These estimates are valid with constants independent of the plate thickness. Finally, we report several numerical experiments showing the behavior of the methods.

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

In the reliability theory, the availability ofa component, characterized by non constant failure and repair rates,is obtained, at a given time, thanks to the computation of the marginal distributions of asemi-Markov process. These measures are shown to satisfy classicaltransport equations, the approximation of which can be donethanks to a finite volume method.Within a uniqueness result for the continuous solution,the convergence of the numerical scheme isthen proven in the weak measure sense,and some numerical applications, which show the efficiency and theaccuracy of the method, are given.

A Volterra model with mutual interferenceconcerning integrated pest management is proposed and analyzed. Byusing Floquet theorem and small amplitude perturbation method andcomparison theorem, we show the existence of a globallyasymptotically stable pest-eradication periodic solution. Further,we prove that when the stability of pest-eradication periodicsolution is lost, the system is permanent and there exists alocally stable positive periodic solution which arises from thepest-eradication periodic solution by bifurcation theory. When theunique positive periodic solution loses its stability, numericalsimulation shows there is a characteristic sequence ofbifurcations, leading to a chaotic dynamics. Finally, we comparethe validity of integrated pest management (IPM) strategy withclassical methods and conclude IPM strategy is more effective thanclassical methods.