Lagrange非结构网格高阶交错型守恒气体动力学格式

提出Lagrange(拉格朗日)非结构网格高阶交错型守恒气体动力学格式.用产生于当前时刻子网格密度和网格声速的子网格压力和MUSCL方法构造了高阶子网格力,利用高阶子网格力构造了高阶空间通量,借助时间中点通量的Taylor(泰勒)展开完成了高阶时间通量离散.研制了Lagrange非结构网格高阶交错型守恒气体动力学格式.对Saltzman活塞问题等进行了数值模拟,数值结果显示了Lagrange非结构网格高阶交错型守恒气体动力学格式的有效性和精确性.     

Lagrange柱坐标高阶中心型守恒格式

提出Lagrange柱坐标高阶中心型守恒格式.基于用对守恒律的单调迎风算法(MUSCL)构造的高阶子网格压力,引入了柱坐标高阶体权子网格力和柱坐标高阶面权子网格力,构造了柱坐标高阶体权中心型守恒格式和柱坐标高阶面权中心型格式.柱坐标高阶体权中心型守恒格式满足动量守恒、能量守恒,但不能确定保持一维球对称性.柱坐标高阶面权中心型格式满足能量守恒,保持一维球对称性.两种格式里,格点速度以与网格面的数值通量相容的方式计算.对Saltzman活塞问题等进行了数值模拟,数值结果显示Lagrange柱坐标高阶中心型守恒格式的有效性和精确性.     

非线性抛物型方程组的二次有限体积元方法及其误差估计

讨论基于三角形网格的二维非线性抛物型方程组的有限体积元方法,其中试探函数空间为二次Lagrange元,检验函数空间为分片常数函数空间,对问题的全离散格式证明了最优的能量模误差估计。最后给出一个相关数值算例以验证格式的有效性。     

一种新的求解双曲型守恒律的三阶中心差分格式

本文提出了一种求解双曲型守恒律新的三阶中心差分格式，主要是引入了一种推广的三阶重构，并证明了这种重构在网格边界无振荡．所提的格式保持了中心差分格式简单的优点，不需用Riemann解算器，避免了进行特征解耦．数值试验结果表明本文格式是高精度、高分辨率的。     

二维热传导型半导体瞬态问题的二次元特征有限体积元方法

将特征线方法与有限体积元方法相结合,采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间构造了二维热传导型半导体瞬态问题的全离散二次元特征有限体积元格式,并进行误差分析,得到了次优阶L^2模误差估计结果．     

扩散方程保正的有限体积格式

在大变形网格上数值求解多介质扩散方程时, 如何构造具有保正性的扩散格式一直是人们关注的难题. 本文将简要综述与保正性相关的扩散格式的研究历史, 并为解决这一难题提出新的设计途径,构造出新的具有较高精度的单元中心型守恒保正格式, 它们可兼顾网格几何变形和物理量变化. 本文将给出数值实验结果, 验证新格式在变形的网格上保持非负性.     

热传导型半导体器件的二次元特征有限体积元方法及分析：H^1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合,采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶 H1模误差估计结果.     

一维单个守恒型方程的二阶熵耗散格式

本文考虑一维单个守恒律方程，对其设计了一种非线性守恒型差分格式，此格式为二阶Godunov型的，用的是分片线性重构，重构函数的斜率是根据熵耗散得到的，格式满足熵条件，且数值实验表明格式具有非线性稳定性，在此格式中一个所谓的熵耗散函数起了很重要的作用，它在每个网格的计算中耗散熵，在文中我们给出了熵耗散函数应满足的条件，并给出了一种具体的构造形式，最后给出了一些数值算例，从中可看出熵耗散函数是如何抑制非物理振荡的，及格式对计算的有效性。     

对流扩散方程带罚函数项的有限体积格式及其分析

1引言 有限体积方法[l]一l’]作为守恒型的离散技术，被广泛应用于工程计算领域.文【2，3} 基于分片常数和分片常向量函数空间，对二维驻定对流扩散方程提出了一类非协调混合 有限体积(Covolume)格式，证明了格式具有。(hl/2)收敛精度.但该格式要求对偶剖分 比较规则，即采用重     

热传导型半导体器件的二次元特征有限体积元方法及分析:H1模估计

该文构造了热传导型半导体器件的全离散特征有限体积元格式,将特征线方法与有限体积元方法相结合, 采用Lagrange型分片二次多项式空间和分片常数函数空间分别作为试探函数和检验函数空间,并进行误差分析,得到了最优阶H¹模误差估计结果.     

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.     

A cell-centered lagrangian scheme in two-dimensional cylindrical geometry

A new Lagrangian cell-centered scheme for two-dimensional compressible flows in planar geometry is proposed by Maire et al.The main new feature of the algorithm is that the vertex velocities and the numerical fluxes through the cell interfaces are all evaluated in a coherent manner contrary to standard approaches.In this paper the method introduced by Maire et al.is extended for the equations of Lagrangian gas dynamics in cylindrical symmetry.Two different schemes are proposed,whose difference is that one uses volume weighting and the other area weighting in the discretization of the momentum equation.In the both schemes the conservation of total energy is ensured,and the nodal solver is adopted which has the same formulation as that in Cartesian coordinates.The volume weighting scheme preserves the momentum conservation and the area-weighting scheme preserves spherical symmetry.The numerical examples demonstrate our theoretical considerations and the robustness of the new method.     

Stability of a Cartesian grid projection method for zero Froude number shallow water flows

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov–Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test functions. The key innovation compared to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection. The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method’s applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme.     

A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

We investigate the Cauchy problem for the inhomogeneous Navier‐Stokes equations in the whole n‐dimensional space. Under some smallness assumption on the data, we show the existence of global‐in‐time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p‐1}_{p,1}({\Bbb R}^n)$ . In particular, piecewise‐constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.     

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.     

Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.     

A virtual element method-based positivity-preserving conservative scheme for convection–diffusion problems on polygonal meshes

In this paper, we propose a positivity-preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity-preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell-vertex values by the lowest-order VEM. Then, the nonlinear two-point flux approximations are utilized to obtain the nonnegativity of cell-centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two-point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second-order accuracy for the solution and the first-order accuracy for its gradient. Moreover, the obtained cell-centered values are nonnegative, which demonstrates the positivity-preserving property of our new scheme.     

Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.

     

A deterministic Lagrangian particle separation-based method for advective-diffusion problems

A simple and robust Lagrangian particle scheme is proposed to solve the advective-diffusion transport problem. The scheme is based on relative diffusion concepts and simulates diffusion by regulating particle separation. This new approach generates a deterministic result and requires far less number of particles than the random walk method. For the advection process, particles are simply moved according to their velocity. The general scheme is mass conservative and is free from numerical diffusion. It can be applied to a wide variety of advective-diffusion problems, but is particularly suited for ecological and water quality modelling when definition of particle attributes (e.g., cell status for modelling algal blooms or red tides) is a necessity. The basic derivation, numerical stability and practical implementation of the NEighborhood Separation Technique (NEST) are presented. The accuracy of the method is demonstrated through a series of test cases which embrace realistic features of coastal environmental transport problems. Two field application examples on the tidal flushing of a fish farm and the dynamics of vertically migrating marine algae are also presented.     

Nonconservative scheme with the isentropic condition in rarefaction waves as applied to the compressible Euler equations

A previously developed second-order accurate quasi-monotone scheme is tested using the Riemann problem with high initial pressure and density ratios. For shock waves, the scheme is conservative, while, in rarefaction waves, the isentropic condition along the trajectory of a Lagrangian particle is used instead of conservativeness in energy. It is shown that the shock front position produced by the scheme has no considerable errors typical of a representative set of conservative quasi-monotone schemes of various orders of accuracy. The numerical accuracy is significantly improved in the case of moving grids with a contact discontinuity explicitly introduced in the form of a grid node. It is shown how the method can be extended to cover the multidimensional case and the presence of additional terms in the original equations.