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Science china Mathematics 2020年第1期63卷 183-204页

作者： Haijin Wang Qiang Zhang Shiping Wang chi-Wang shu School of Science Nanjing University of Posts and TelecommunicationsNanjing 210023China Department of Mathematics Nanjing UniversityNanjing 210093China College of Shipbuilding Engineering Harbin Engineering UniversityHarbin 150001China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)*** basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx *** give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order *** optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.

关键词： local discontinuous Galerkin explicit-implicit-null time discretization nonlinear diffusion stability error estimates

A New Approach of High OrderWell-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

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Communications in Computational Physics 2006年第1期1卷 100-134页

作者： Yulong Xing chi-Wang shu Department of Mathematics Brown UniversityProvidenceRI 02912USA Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source *** our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source *** this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element *** make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is *** computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG *** same idea can be applied to the finite volume WENO *** will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance *** perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.

关键词： Hyperbolic balance laws WENO finite volume scheme discontinuous Galerkin method high order accuracy source term conservation laws shallow water equation elastic wave equation chemosensitive movement nozzle flow two phase flow

Development of Residual Distribution Schemes for the Discontinuous Galerkin Method:The Scalar Case with Linear Elements

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Communications in Computational Physics 2009年第2期5卷 376-390页

作者： Remi Abgrall chi-Wang shu Institut de Mathematiques and INRIA Universite Bordeaux I341 cours de la Liberation33405 Talence cedexFrance Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper,we reformulate the piecewise linear discontinuous Galerkin(DG)method for solving two dimensional steady state scalar conservation laws in the framework of residual distribution(RD)*** allows us to propose a new class of nonlinear stabilization that does not destroy the formal accuracy of the *** results are shown to demonstrate the behavior of this approach.

关键词： Discontinuous Galerkin schemes residual distribution schemes high order unstructured meshes

Mechanical properties of defective L1_(2)-Al_(3)X(X=Sc,Lu)phase:A first-principles study

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Journal of Rare Earths 2021年第2期39卷 217-224页

作者： Yu Wang Ya'nan meng Junsheng Wang chi Zhang Houbing Huang School of Materials Science and Engineering Beijing Institute of TechnologyBeijing 100081China Advanced Research Institute of Multidisciplinary Science Beijing Institute of TechnologyBeijing 100081China

The mechanical properties of Al_(3)X(X=Sc,Lu)were studied by density functional theory(DFT).The elastic constants and formation enthalpy indicate that the L1_(2)-Al_(3)X(X=Sc and Lu)are mechanically and thermodynamically *** bulk moduli and shear moduli show that Al_(3)Sc has better resistance to volume and shape changes than AI3 ***,the calculated results show that Al_(3)Lu has better plasticity than Al_(3)*** properties of structural stability and elastic moduli of the crystal containing four major types of point defects in L1_(2)-Al_(3)X(X=Sc and Lu)were *** mechanical properties of point defects show that point defects cause L1_(2)-Al_(3)X lattice distortion and change the corresponding elastic *** defects reduce the Young’s,shear and bulk moduli but have little effects on the crystal brittleness and toughness of Al_(3)Sc and Al_(3)***,we have found that Lu addition into aluminum alloys is a very good replacement for expensive Sc addition when the L1_(2)structures are desired for nucleation or strengthening precipitates in aluminum alloys.

关键词： Density functional theory First-principle L1_(2)-Al_(3)X Mechanical property Defects Rare earths

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

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Communications on Applied Mathematics and Computation 2019年第1期1卷 101-116页

作者： Yong Liu chi-Wang shu mengping Zhang School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper,we study the superconvergence properties of the energy-conserving discontinuous Galerkin(DG)method in[18]for one-dimensional linear hyperbolic *** prove the approximate solution superconverges to a particular projection of the exact *** order of this superconvergence is proved to be k+2 when piecewise Pk polynomials with K≥1 are *** proof is valid for arbitrary non-uniform regular meshes and for piecewise polynomials with arbitrary K≥***,we find that the derivative and function value approxi?mations of the DG solution are superconvergent at a class of special points,with an order of k+1 and R+2,*** also prove,under suitable choice of initial discretization,a(2k+l)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell *** experiments are given to demonstrate these theoretical results.

关键词： Energy-conserving discontinuous Galerkin methods Superconvergence Linear hyperbolic equations

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

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Communications in Computational Physics 2022年第7期32卷 364-400页

作者： Jiayin Li chi-Wang shu Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathe-matical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation *** main idea of this scheme is derived from our previous work[***.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the ***,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order *** addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near *** obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO ***,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO *** numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

关键词： Moment-based scheme multi-resolution scheme HWENO scheme hyperbolic conservation laws KXRCF troubled-cell indicator HLLC-flux

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

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Communications in Computational Physics 2012年第10期12卷 1307-1328页

作者： Juan Cheng chi-Wang shu Qinghong Zeng National Key Laboratory of Science and Technology on Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA Institute of Applied Physics and Computational Mathematics Beijing 100088China

Lagrangianmethods arewidely used inmany fields formulti-material compressible flow simulations such as in astrophysics and inertial confinement fusion(ICF),due to their distinguished advantage in capturing material interfaces *** some of these applications,multiple internal energy equations such as those for electron,ion and radiation are *** the past decades,several staggeredgrid based Lagrangian schemes have been developed which are designed to solve the internal energy equation *** schemes can be easily extended to solve problems with multiple internal energy *** such schemes are typically not conservative for the total ***,significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for ***,these schemes are commonly designed to solve the Euler equations in the form of the total energy,therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant *** modifications,if not designed carefully,may lead to the loss of some of the nice properties of the original schemes such as conservation of the total *** this paper,we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal *** a carefully designed modification in the implementation,the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation *** advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy *** two dimensional numerical examp

关键词： Lagrangian scheme conservative cell-centered internal energy equation compressible fluid flow three-temperaturemodel

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

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Science china Mathematics 2013年第12期56卷 2645-2676页

作者： LU JianFang shu chi-Wang ZHANG mengPing School of Mathematical Sciences University of Science and Technology of China Division of Applied Mathematics Brown UniversityProvidence RI 02912 USA

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in *** RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical *** using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth *** theoretical results are proved for piecewise polynomials with any degree k ***,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising *** results are provided to demonstrate these RKDG methods.

关键词： discontinuous Galerkin method explicit Runge-Kutta method stability error estimates corre-lated random walk positivity-preserving

On the Order of Accuracy and Numerical Performance of Two Classes of Finite Volume WENO Schemes

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Communications in Computational Physics 2011年第3期9卷 807-827页

作者： Rui Zhang mengping Zhang chi-Wang shu Department of Mathematics University of Science and Technology of ChinaHefeiAnhui 230026China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory(WENO)schemes in two dimensional Cartesian *** compare them in terms of accuracy,performance for smooth and shocked solutions,and efficiency in CPU *** linear systems both schemes are high order accurate,however for nonlinear systems,analysis and numerical simulation results verify that one of them(Class A)is only second order accurate,while the other(Class B)is high order *** WENO scheme in Class A is easier to implement and costs less than that in Class *** experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes,despite of the difference in their formal order of *** results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

关键词： Weighted essentially non-oscillatory(WENO)schemes finite volume schemes accuracy

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

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Communications in Computational Physics 2023年第10期34卷 1177-1214页

作者： Xiaolu Gu Juan Cheng chi-Wang shu Graduate School China Academy of Engineering PhysicsBeijing 100088China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China HEDPS Center for Applied Physics and Technologyand College of EngineeringPeking UniversityBeijing 100871China Division of Applied Mathematics Brown UniversityProvidenceRI 02912

The arbitrary Lagrangian-Eulerian(ALE)method is widely used in the field of compressible multi-material and multi-phase flow *** order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin(DG)discretizations,we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this *** remapping method is based on the ALE-DG method developed by Klingenberg et al.[17,18]to solve the trivial equation∂u/∂t=0 on a moving mesh,which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=*** appropriate selection of the final pseudo-time T can always satisfy the relatively mild smoothness requirement(Lipschitz continuity)on the mesh movement velocity,which guarantees the high order accuracy of the remapping *** use a multi-resolution weighted essentially non-oscillatory(WENO)limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth *** further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order *** experiments are provided to illustrate the high order accuracy,essentially non-oscillatory performance and positivity-preserving of our remapping *** addition,the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one-and two-dimensional Euler equations.

关键词： Remapping discontinuous Galerkin method arbitrary Lagrangian-Eulerian high order accuracy multi-resolution WENO limiter positivity-preserving