A non-dissipative entropic scheme for convex scalar equations via discontinuous cell-reconstruction

We present a finite volume non-dissipative but entropic scheme for convex scalar equations based on a discontinuous reconstruction of the solution in each cell of the mesh. This discontinuous representation of the numerical solution in each cell is done satisfying the L^∞-norm, Total Variation and entropy decreasing properties. This allows us to prove the convergence towards the unique entropy solution. Numerical computations are reported, showing the non-dissipative behavior of the algorithm. To cite this article: F. Lagoutière, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution of axisymmetric Maxwell equations

In this article, we study the static and time‐dependent Maxwell equations in axisymmetric geometry. Using the mathematical tools introduced in (Math. Meth. Appl. Sci. 2002; 25 : 49), we investigate the decoupled problems induced in a meridian half‐plane, and the splitting of the solution in a regular part and a singular part, the former being in the Sobolev space H¹ component‐wise. It is proven that the singular parts are related to singularities of Laplace‐like or wave‐like operators. We infer from these characterizations: (i) the finite dimension of the space of singular fields; (ii) global space and space–time regularity results for the electromagnetic field. This paper is the continuation of (Modél. Math. Anal. Numér. 1998; 32 : 359, Math. Meth. Appl. Sci. 2002; 25 : 49). Copyright © 2003 John Wiley & Sons, Ltd.     

An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation

We extend a theorem of Jörgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampère equation to parabolic Monge–Ampère equation, and obtain delicate asymptotic behavior of solutions at infinity. For the dimension \(n\ge 3\), the work of Gutiérrez and Huang in Indiana Univ. Math. J. 47, 1459–1480 (1998) is an easy consequence of our result. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations.     

Multiplicité des solutions d'équilibre d'une coque de Marguerre–von Kármán membranaireMultiplicity of solutions to the Marguerre–von Kármán membrane equations

We build explicitly an infinite number of equilibrium solutions of unloaded Marguerre–von Kármán membrane shells. This construction is based upon the existence of three elementary solutions, together with the solution of a Monge–Ampère equation associated with a partition of the reference configuration of the shell. To cite this article: A. Léger, B. Miara, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 649–654.     

A high‐order ADI finite difference scheme for a 3D reaction‐diffusion equation with neumann boundary condition

In this article, we extend the fourth‐order compact boundary scheme in Liao et al. (Numer Methods Partial Differential Equations 18 (2002), 340–354) to a 3D problem and then combine it with the fourth‐order compact alternating direction implicit (ADI) method in Gu et al. (J Comput Appl Math 155 (2003), 1–17) to solve the 3D reaction‐diffusion equation with Neumann boundary condition. First, the reaction‐diffusion equation is solved with a compact fourth‐order finite difference method based on the Padé approximation, which is then combined with the ADI method and a fourth‐order compact scheme to approximate the Neumann boundary condition, to obtain fourth order accuracy in space. The accuracy in the temporal dimension is improved to fourth order by applying the Richardson extrapolation technique, although the unconditional stability of the numerical method is proved, and several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed new algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties

Abstract

Following Contou-Carrère (Contou-Carrère,C. (1983). Géométrie des Groupes Semi-Simples,Résolutions équivariantes et Lieu Singulier de Leurs Variétés de Schubert. Thèse d’état,Université Montpellier II (published partly as,Le Lieu singulier des variétés de Schubert (1988). Adv. Math.,71:186–221)),we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. Using Carrell and Peterson's characterization (Carrell,J. B. (1994). The Bruhat graph of a Coxeter group,a conjecture of Deodhar,and rational smoothness of Schubert varieties. Proc. Symp. in Pure Math. 56(Part I):53–61),we prove that rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called,the combinatorial tangent space.     

A finite volume method on general meshes for a degenerate parabolic convection–reaction–diffusion equation

We propose a finite volume method on general meshes for the discretization of a degenerate parabolic convection–reaction–diffusion equation. Equations of this type arise in many contexts, such as for example the modeling of contaminant transport in porous media. The diffusion term, which can be anisotropic and heterogeneous, is discretized using a recently developed hybrid mimetic mixed framework. We construct a family of discretizations for the convection term, which uses the hybrid interface unknowns. We consider a wide range of unstructured possibly nonmatching polyhedral meshes in arbitrary space dimension. The scheme is fully implicit in time, it is locally conservative and robust with respect to the Péclet number. We obtain a convergence result based upon a priori estimates and the Fréchet–Kolmogorov compactness theorem. We implement the scheme both in two and three space dimensions and compare the numerical results obtained with the upwind and the centered discretizations of the convection term numerically.     

Réalisation de Hodge des motifs de Voevodsky

Sur un sous-corps du corps des complexes, nous associons à un objet de la catégorie des motifs géométriques de Voevodsky un Z-complexe de Hodge mixte de Deligne (Inst Hautes études Sci Publ Math 44:5–77, [D74]). Les filtrations par le poids et de Hodge sont représentées par des foncteurs de troncature, d’un complexe des poids à la Bondarko (J K-Theory 6(3):387–504, [Bo10a]) pour la première et troncature du complexe de De Rham (Lecomte Wach, Manuscr Math 129:75–90, [LW09]) pour l’autre.     

A new fifth order finite difference WENO scheme for Hamilton‐Jacobi equations

In this continuing paper of (Zhu and Qiu, J Comput Phys 318 (2016), 110–121), a new fifth order finite difference weighted essentially non‐oscillatory (WENO) scheme is designed to approximate the viscosity numerical solution of the Hamilton‐Jacobi equations. This new WENO scheme uses the same numbers of spatial nodes as the classical fifth order WENO scheme which is proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143), and could get less absolute truncation errors and obtain the same order of accuracy in smooth region simultaneously avoiding spurious oscillations nearby discontinuities. Such new WENO scheme is a convex combination of a fourth degree accurate polynomial and two linear polynomials in a WENO type fashion in the spatial reconstruction procedures. The linear weights of three polynomials are artificially set to be any random positive constants with a minor restriction and the new nonlinear weights are proposed for the sake of keeping the accuracy of the scheme in smooth region, avoiding spurious oscillations and keeping sharp discontinuous transitions in nonsmooth region simultaneously. The main advantages of such new WENO scheme comparing with the classical WENO scheme proposed by Jiang and Peng (SIAM J Sci Comput 21 (2000), 2126–2143) are its efficiency, robustness and easy implementation to higher dimensions. Extensive numerical tests are performed to illustrate the capability of the new fifth WENO scheme. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1095–1113, 2017     

A convergent three‐level finite difference scheme for solving a dual‐phase‐lagging heat transport equation in spherical coordinates

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three‐level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60–71, 2004.     

Development of a numerical phase optimized upwinding combined compact difference scheme for solving the Camassa–Holm equation with different initial solitary waves

In this article, the solution of Camassa–Holm (CH) equation is solved by the proposed two‐step method. In the first step, the sixth‐order spatially accurate upwinding combined compact difference scheme with minimized phase error is developed in a stencil of four points to approximate the first‐order derivative term. For the purpose of retaining both of the long‐term accurate Hamiltonian property and the geometric structure inherited in the CH equation, the time integrator used in this study should be able to conserve symplecticity. In the second step, the Helmholtz equation governing the pressure‐like variable is approximated by the sixth‐order accurate three‐point centered compact difference scheme. Through the fundamental and numerical verification studies, the integrity of the proposed high‐order scheme is demonstrated. Another aim of this study is to reveal the wave propagation nature for the investigated shallow water equation subject to different initial wave profiles, whose peaks take the smooth, peakon, and cuspon forms. The transport phenomena for the cases with/without inclusion of the linear first‐order advection term κu_x in the CH equation will be addressed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1645–1664, 2015     

Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels

This work is concerned with stabilization of a wave equation stabilized by a boundary feedback. When the feedback is both frictional and with memory, we prove exponential stability of the solutions. In case of a boundary feedback which is only of memory type, uniform stability is not expected. We prove in this latter case, that the solutions decay polynomially. The method is new and uses the method of higher order energies (see [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]), the multiplier method and the properties of a large class of singular kernels. Moreover, our method can be extended to include cases of nonsingular kernels (see [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]). To cite this article: F. Alabau-Boussouira et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Analytical and numerical stability of partial differential equations with piecewise constant arguments

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014     

On the solution of Dirichlet problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the second type

Complex Monge-Ampère equation is a nonlinear equation with high degree, so its solution is very difficult to get. How to get the plurisubharmonic solution of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain of the second type is discussed by using the analytic method in this paper. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear second-order ordinary differential equation (ODE) by using quite different method. Secondly, the solution of the Dirichlet problem is given in semi-explicit formula, and under a special case the exact solution is obtained. These results may be helpful for the numerical method of Dirichlet problem of complex Monge-Ampère equation on the Cartan-Hartogs domain.     

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

In this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.     

Multi-frame representations in linear inverse problems with mixed multi-constraints

This paper is concerned with linear inverse problems where the solution is assumed to have a sparse expansion with respect to several bases or frames. We were mainly motivated by the following two different approaches: (1) Jaillet and Torrésani [F. Jaillet, B. Torrésani, Time–frequency jigsaw puzzle: Adaptive multi-window and multi-layered Gabor expansions, preprint, 2005] and Molla and Torrésani [S. Molla, B. Torrésani, A hybrid audio scheme using hidden Markov models of waveforms, Appl. Comput. Harmon. Anal. (2005), in press] have suggested to represent audio signals by means of at least a wavelet for transient and a local cosine dictionary for tonal components. The suggested technology produces sparse representations of audio signals that are very efficient in audio coding. (2) Also quite recently, Daubechies et al. [I. Daubechies, M. Defrise, C. DeMol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. 57 (2004) 1413–1541] have developed an iterative method for linear inverse problems that promote a sparse representation for the solution to be reconstructed. Here in this paper, we bring both ideas together and construct schemes for linear inverse problems where the solution might then have a sparse representation (we also allow smoothness constraints) with respect to several bases or frames. By a few numerical examples in the field of audio and image processing we show that the resulting method works quite nicely.     

A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence

We introduce a two-grid finite difference approximation scheme for the free Schrödinger equation. This scheme is shown to converge and to posses appropriate dispersive properties as the mesh-size tends to zero. A careful analysis of the Fourier symbol shows that this occurs because the two-grid algorithm (consisting in projecting slowly oscillating data into a fine grid) acts, to some extent, as a filtering one. We show that this scheme converges also in a class of nonlinear Schrödinger equations whose well-posedness analysis requires the so-called Strichartz estimates. This method provides an alternative to the method introduced by the authors [L.I. Ignat, E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris 340 (7) (2005) 529–534] using numerical viscosity. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Parallel solution in time of ODEs: some achievements and perspectives

The parallel solution of initial value problems for ordinary differential equations (ODE-IVPs) has received much interest from many researchers in the past years. In general, the possibility of using parallel computing in this setting concerns different aspects of the numerical solution of ODEs, depending on the parallel platform to be used and/or the complexity of the problem to be solved. In particular, in this paper we examine possible extensions of a parallel method previously proposed in the mid-nineties [P. Amodio, L. Brugnano, Parallel implementation of block boundary value methods for ODEs, J. Comput. Appl. Math. 78 (1997) 197–211; P. Amodio, L. Brugnano, Parallel ODE solvers based on block BVMs, Adv. Comput. Math. 7 (1997) 5–26], and analyze its connections with subsequent approaches to the parallel solution of ODE-IVPs, in particular the “Parareal” algorithm proposed in [J.L. Lions, Y. Maday, G. Turinici, Résolution d'EDP par un schéma en temps “pararéel”, C. R. Acad. Sci. Paris, Ser. I 332 (2001) 661–668; Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 387–392].     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.