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High Order Compact Schemes in Projection Methods for Incompressible Viscous Flows

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Communications in Computational Physics 2011年第4期9卷 994-1019页

作者： Michel Fournie Alain Rigal institut de mathematiques de toulouse Universite de Toulouse et CNRS(UMR 5219)France

Within the projection schemes for the incompressible Navier-Stokes equations(namely"pressure-correction"method),we consider the simplest method(of order one in time)which takes into account the pressure in both steps of the splitting *** this scheme,we construct,analyze and implement a new high order compact spatial approximation on nonstaggered *** approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions(without error on the velocity)which could be extended to more general *** prove the unconditional stability of the associated Cauchy problem via von Neumann *** we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical *** we present detailed numerical tests for the Stokes and the Navier-Stokes equations(including the driven cavity benchmark)to illustrate the theoretical results.

关键词： Incompressible Navier-Stokes fractional step method high order compact scheme boundary conditions

degenerate Anisotropic Elliptic Problems and Magnetized Plasma Simulations

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Communications in Computational Physics 2012年第1期11卷 147-178页

作者： Stephane Brull Pierre degond Fabrice deluzet institut de mathematiques de Bordeaux UMR 5251 Equipe Mathematiques Appliquees de Bordeaux(MAB)Universite Bordeaux 1351cours de la Liberation-33405 TALENCE cedexFrance Universite de toulouse UPSINSAUT1UTMInstitut de Mathematiques de ToulouseF-31062 ToulouseFrance CNRS Institut de Mathematiques de Toulouse UMR 5219F-31062 ToulouseFrance

This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic *** numerical method is designed for arbitrary spacedependent anisotropy directions and does not require any specially adapted coordinate *** is also designed to be equally accurate in the strongly and the mildly anisotropic *** method is applied to the Euler-Lorentz system,in the drift-fluid *** system provides a model for magnetized plasmas.

关键词： Anisotropic elliptic problem asymptotic-preserving scheme Lorentz force large magnetic field low-Mach number drift-fluid limit

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep, Circulating Flow, and Pumping Effect

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Communications in Computational Physics 2009年第10期6卷 919-954页

作者： Kazuo Aoki Pierre degond Luc Mieussens department of Mechanical Engineering and Science Graduate School of EngineeringKyoto UniversityKyoto 606-8501Japan Universite de toulouse UPSINSAUT1UTMInstitut de Mathematiques de ToulouseF-31062 ToulouseFrance CNRS Institut de Mathematiques de Toulouse UMR 5219F-31062 ToulouseFrance Universite de Bordeaux Institut de Mathematiques de Bordeaux351cours de la Liberation33405 Talence cedexFrance

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of *** device is made of a simple smooth and curved channel with a periodic temperature *** the Boltzmann-BGK model as the governing equation for the gas flow,we develop a numerical method based on a deterministic finite volume scheme,implicit in time,with an implicit treatment of the boundary *** method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices,and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the *** simulations show the ability of the device to produce a one-way flow that has a pumping effect.

关键词： Knudsen compressor thermal creep flow transpiration flow BGK model implicit scheme

Bounded critical Fatou components are Jordan domains for polynomials

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Science China Mathematics 2022年第2期65卷 331-358页

作者： Pascale Roesch Yongcheng Yin institut de mathematiques de toulouse Universite Paul SabatierToulouse 31062France School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China

We prove that any bounded Fatou component of a polynomial of degree at least two, which is not(eventually) a Siegel disk, is a Jordan domain.

关键词： Fatou component Jordan domain puzzle local connectivity

Hamilton-type Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds

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Acta Mathematica Sinica,English Series 2011年第6期27卷 1071-1078页

作者： Bin QIAN department of Mathematics Changshu Institute of Technology Changshu 215500 P. R. China institut de mathematiques de toulouse Universitd de Toulouse CNRS 5219 France

Let （M, g） be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equationon tu=△u＋aulogu＋qu on M × （0, ∞）, where a is a constant and q is a C2 function. This result can be compared with the ones of Ma （JFA, 241, 374-382 （2006）） and Yang （PAMS, 136, 4095-4102 （2008））. Also, we obtain Hamilton＇s gradient estimate for the Schodinger equation. This can be compared with the result of Ruan （JGP, 58, 962-966 （2008））.

关键词： Nonlinear parabolic equations Li-Yau inequalities Harnack differential inequalities gradient estimates

Bounded type Siegel disks of finite type maps with few singular values dedicated to the Memory of Professor Lei Tan

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Science China Mathematics 2018年第12期61卷 2139-2156页

作者： Arnaud Chéritat Adam Lawrence Epstein CNRS Institut de Mathematiques de ToulouseUniversite Paul SabatierToulouse F-31062France Mathematics institute University of WarwickCoventry CV47ALUK

Let U be an open subset of the Riemann sphere C. We give sufficient conditions for which a finite type map f : U →C with at most three singular values has a Siegel disk compactly contained in U and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery à la Douady-Ghys-Herman-(S′)wia tek. We also give sufficient conditions for which, instead, ? has not compact closure in U. The main tool is the Schwarzian derivative and area inequalities à la Graczyk-(S′)wia tek.

关键词： Siegel disks quasicircles quasiconformal surgery Schwarzian derivative

Thermal Convection in a Spherical Shell with Melting/Freezing at either or both of Its Boundaries

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Journal of Earth Science 2013年第5期24卷 669-682页

作者： Renaud deguen institut de Mécanique des Fluides de toulouse Université de Toulouse (INPT UPS) and CNRS

In a number of geophysical or planetological settings, including Earth＇s inner core, a sili- cate mantle crystallizing from a magma ocean, or an ice shell surrounding a deep water ocean--a situa- tion possibly encountered in a number of Jupiter and Saturn＇s icy satellites--a convecting crystalline layer is in contact with a layer of its melt. Allowing for melting/freezing at one or both of the boundaries of the solid layer is likely to affect the pattern of convection in the layer. We study here the onset of thermal convection in a viscous spherical shell with dynamically induced melting/freezing at either or both of its boundaries. It is shown that the behavior of each interface---permeable or impermeable-- depends on the value of a dimensionless number P （one for each boundary）, which is the ratio of a melting/freezing timescale over a viscous relaxation timescale. A small value of P corresponds to perme- able boundary conditions, while a large value of P corresponds to impermeable boundary conditions. Linear stability analysis predicts a significant effect of semi-permeable boundaries when the number P characterizing either of the boundary is small enough： allowing for melting/freezing at either of the boundary allows the emergence of larger scale convective modes. The effect is particularly drastic when the outer boundary is permeable, since the degree 1 mode remains the most unstable even in the case of thin spherical shells. In the case of a spherical shell with permeable inner and outer boundaries, the most unstable mode consists in a global translation of the solid shell, with no deformation. In the limit of a full sphere with permeable outer boundary, this corresponds to the ＂convective translation＂ mode recently proposed for Earth＇s inner core. As another example of possible application, we discuss the case of thermal convection in Enceladus＇ ice shell assuming the presence of a global subsurface ocean, and found that melting/freezing could have an importa

关键词： planetary mantle thermal convection phase change linear stability analysis.

ELLIPTIC GRADIENT ESTIMATES FOR DIFFUSION OPERATORS ON COMPLETE RIEMANNIAN MANIFOLDS

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Acta Mathematica Scientia 2010年第5期30卷 1555-1560页

作者：钱斌 School of Mathematics and Statistics Wuhan University institut de Math'ematiques de toulouse Universit'e de Toulouse

In this note, we obtain the elliptic estimate for diffusion operator L = △＋△Ф·△ on complete, noncompact Riemannian manifolds, under the curvature condition CD（K, m）, which generalizes B. L. Kotschwar＇s work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for SchrSdinger-type gradient is also derived.

关键词： gradient estimate Bakry-Emery curvature diffusion operator

High Order Accurate Direct Arbitrary-Lagrangian-Eulerian AdeR-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms

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Communications in Computational Physics 2017年第1期21卷 271-312页

作者： Walter Boscheri Raphael Loubere Laboratory of AppliedMathematics Department of CivilEnvironmental and Mechanical EngineeringUniversity of TrentoI-38123 TrentoItaly CNRS and institut de mathematiques de toulouse(IMT)Universite Paul-Sabatier ToulouseFrance

In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source *** scheme is constructed with a single stencil polynomial reconstruction operator,a one-step space-time AdeR integration which is suitably designed for dealing even with stiff sources,a nodal solver with relaxation to determine the mesh motion,a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order detection(MOOD)*** this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become *** new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence *** Riemann problems and explosion problems with exact solutions are simulated in 2D and *** overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

关键词： Direct Arbitrary-Lagrangian-Eulerian a posteriori MOOD stabilization Baer-Nunziato model stiff source terms non-conservative products unstructured mesh AdeR high order of accuracy in space and time high performance computing(HPC) hyperbolic conservation laws