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Journal of Computational Mathematics 2020年第4期38卷 661-682页

作者： Sergio Amat Juan Ruiz chi-wang shu Department of Applied Mathematics and Statistics Universidad Politecnica de CartagenaSpain Division of Applied Mathematics Brown UniversityUSA

This paper is the second part of the article and is devoted to the construction and analysis of new non-linear optimal weights for WENO interpolation capable of rising the order of accuracy close to discontinuities for data discretized in the cell ***,now we are interested in analyze the capabilities of the new algorithm when working with functions belonging to the subspace L1\L2 and that,consequently,are piecewise smooth and can present jump *** new non-linear optimal weights are redesigned in a way that leads to optimal theoretical accuracy close to the discontinuities and at smooth *** will present the new algorithm for the approximation case and we will analyze its *** we will explain how to use the new algorithm in multiresolution applications for univariate and bivariate *** numerical results confirm the theoretical proofs presented.

关键词： WENO Cell averages New optimal weights Multiresolution schemes Im-proved adaption to discontinuities Signal processing

SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

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Journal of Computational Mathematics 2021年第4期39卷 518-537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 theoretically and numerically verify that the discontinuous Galerkin(DG)methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L^(2)-norm for even degree polynomial *** uniform meshes,the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions,improving previous *** theoretical findings are found to be sharp and consistent with numerical results.

关键词： Discontinuous Galerkin method Central flux Sub-optimal convergence rates

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

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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

Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations

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Communications in Computational Physics 2010年第1期7卷 1-46页

作者： Yan Xu chi-wang shu Department of Mathematics University of Science and Technology of ChinaHefeiAnhui 230026China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.

关键词： Discontinuous Galerkin method local discontinuous Galerkin method numerical flux stability time discretization high order accuracy stability error estimates

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

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Communications in Computational Physics 2012年第4期11卷 1144-1168页

作者： Juan Cheng chi-wang shu Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In[14],Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical *** schemes use a node-based discretization of the numerical *** control volume version has several distinguished properties,including the conservation of mass,momentum and total energy and compatibility with the geometric conservation law(GCL).However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical *** alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry,at the price of sacrificing conservation of *** this paper,we apply the methodology proposed in our recent work[8]to the first order control volume scheme of Maire in[14]to obtain the spherical symmetry *** modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid,andmeanwhile itmaintains its original good properties such as conservation and *** two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry,non-oscillation and robustness properties.

关键词： Control volume Lagrangian scheme spherical symmetry preservation conservative cell-centered compressible flow cylindrical coordinates

A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations

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Communications in Computational Physics 2008年第10期4卷 1008-1024页

作者： Juan Cheng chi-wang shu Institute of Applied Physics and Computational Mathematics Beijing 100088China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

Based on the high order essentially non-oscillatory(ENO)Lagrangian type scheme on quadrilateral meshes presented in our earlier work[3],in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas *** main purpose of this work is to demonstrate our claim in[3]that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges,which restricts the accuracy of the resulting scheme to at most second *** accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear *** examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.

关键词： Lagrangian type scheme high order accuracy conservative scheme curvilinear mesh WENO reconstruction compressible Euler equations

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

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Communications in Computational Physics 2011年第7期10卷 474-508页

作者： Yan Xu chi-wang shu Department of Mathematics University of Science and Technology of ChinaHefei 230026AnhuiChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper,we develop,analyze and test local discontinuous Galerkin(LDG)methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives,and possibly discontinuous or sharp transition *** LDG method has the flexibility for arbitrary h and p *** prove the L2 stability for general *** proof of the total variation stability of the schemes for the piecewise constant P0 case is also *** numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

关键词： Local discontinuous Galerkin method Degasperis-Procesi equation L2 stability total variation stability

A HIGH ORDER ADAPTIVE FINITE ELEMENT METHOD FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

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Journal of Computational Mathematics 2011年第5期29卷 491-500页

作者： Zhengfu Xu Jinchao Xu chi-wang shu Department of Mathematics Michigan State University East Lansing MI ~882~ USA Department of Mathematics Pennsylvania State University University Park PA 16802 USA Division of Applied Mathematics Brown University Providence RI 02912 USA

In this note, we apply the h-adaptive streamline diffusion finite element method with a small mesh-dependent artificial viscosity to solve nonlinear hyperbolic partial differential equations, with the objective of achieving high order accuracy and mesh efficiency. We compute the numerical solution to a steady state Burgers equation and the solution to a converging-diverging nozzle problem. The computational results verify that, by suitably choosing the artificial viscosity coefficient and applying the adaptive strategy based on a posterior error estimate by Johnson et al., an order of N^-3/2 accuracy can be obtained when continuous piecewise linear elements are used, where N is the number of elements.

关键词： Adaptive finite element Nonlinear hyperbolic conservation law.

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

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Communications in Computational Physics 2011年第9期10卷 979-1000页

作者： Jing-Mei Qiu chi-wang shu Department of Mathematical and Computer Science Colorado School of MinesGoldenCO 80401USA Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

In this paper,we propose a new conservative semi-Lagrangian(SL)finite difference(FD)WENO scheme for linear advection equations,which can serve as a base scheme for the Vlasov equation by Strang splitting[4].The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume(FV)WENO scheme[3].However,instead of inputting cell averages and approximate the integral form of the equation in a FV scheme,we input point values and approximate the differential form of equation in a FD spirit,yet retaining very high order(fifth order in our experiment)spatial *** advantage of using point values,rather than cell averages,is to avoid the second order spatial error,due to the shearing in velocity(v)and electrical field(E)over a cell when performing the Strang splitting to the Vlasov *** a result,the proposed scheme has very high spatial accuracy,compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson(VP)*** perform numerical experiments on linear advection,rigid body rotation problem;and on the Landau damping and two-stream instabilities by solving the VP *** comparison,we also apply(1)the conservative SL FD WENO scheme,proposed in[22]for incompressible advection problem,(2)the conservative SL FD WENO scheme proposed in[21]and(3)the non-conservative version of the SL FD WENO scheme in[3]to the same test *** performances of different schemes are compared by the error table,solution resolution of sharp interface,and by tracking the conservation of physical norms,energies and entropies,which should be physically preserved.

关键词： Semi-Lagrangian methods finite difference/finite volume scheme conservative scheme WENO reconstruction Vlasov equation Landau damping two-stream instability

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