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DISSIPATIVE NUMERICAL METHODS FOR THE HUNTER-SAXTON EQUATION

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Journal of Computational Mathematics 2010年第5期28卷 606-620页

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

In this paper, we present further development of the local discontinuous Galerkin （LDG） method designed in [21] and a new dissipative discontinuous Galerkin （DG） method for the HuntermSaxton equation. The numerical fluxes for the LDG and DG methods in this paper are based on the upwinding principle. The resulting schemes provide additional energy dissipation and better control of numerical oscillations near derivative singularities. Stability and convergence of the schemes are proved theoretically, and numerical simulation results are provided to compare with the scheme in [21].

关键词： Discontinuous Galerkin method Local discontinuous Galerkin method dissi-pation Hunter-Saxton equation Stability.

ANALYSIS OF SHARP SUPERCONVERGENCE OF LOCAL DISCONTINUOUS GALERKIN METHOD FOR ONE-DIMENSIONAL LINEAR PARABOLIC EQUATIONS

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Journal of Computational Mathematics 2015年第3期33卷 323-340页

作者： Yang Yang chi-wang shu Department of Mathematical Sciences Michigan Technological University Houghton MI 49931. Division of Applied Mathematics Brown University Providence RI 02912.

In this paper, we study the superconvergence of the error for the local discontinuous Galerkin （LDG） finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is （k ＋ 2）-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is （k ＋ 2）-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for P^k polynomials with arbitrary k ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp.

关键词： Superconvergence Local discontinuous Galerkin method Parabolic equation Lnitial discretization Error estimates Radau points.

STRONG STABILITY PRESERVING PROPERTY OF THE DEFERRED CORRECTION TIME DISCRETIZATION

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Journal of Computational Mathematics 2008年第5期26卷 633-656页

作者： Yuan Liu chi-wang shu Mengping Zhang Department of Mathematics University of Science and Technology of China Hefei 230026 China Division of Applied Mathematics Brown University Providence RI 02912 USA Department of Mathematics University of Science and Technology of China Hefei 230026 China

In this paper, we study the strong stability preserving （SSP） property of a class of deferred correction time discretization methods, for solving the method-of-lines schemes approximating hyperbolic partial differen... 详细信息

关键词： Strong stability preserving Deferred correction time discretization.

NUMERICAL BOUNDARY CONDITIONS FOR THE FAST SWEEPING HIGH ORDER WENO METHODS FOR SOLVING THE EIKONAL EQUATION

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Journal of Computational Mathematics 2008年第3期26卷 336-346页

作者： Ling Huang chi-wang shu Mengping Zhang Department of Mathematics University of Science and Technology of China Hefei Anhui 230026 China Division of Applied Mathematics Brown University Providence RI 02912 USA Department of Mathematics University of Science and Technology of China Hefei 230026 China

High order fast sweeping methods have been developed recently in the literature to solve static Hamilton-Jacobi equations efficiently. Comparing with the first order fast sweeping methods, the high order fast sweeping methods are more accurate, but they often require additional numerical boundary treatment for several grid points near the boundary because of the wider numerical stencil. It is particularly important to treat the points near the inflow boundary accurately, as the information would flow into the computational domain and would affect global accuracy. In the literature, the numerical solution at these boundary points are either fixed with the exact solution, which is not always feasible, or computed with a first order discretization, which could reduce the global accuracy. In this paper, we discuss two strategies to handle the inflow boundary conditions. One is based on the numerical solutions of a first order fast sweeping method with several different mesh sizes near the boundary and a Richardson extrapolation, the other is based on a Lax-Wendroff type procedure to repeatedly utilizing the PDE to write the normal spatial derivatives to the inflow boundary in terms of the tangential derivatives, thereby obtaining high order solution values at the grid points near the inflow boundary. We explore these two approaches using the fast sweeping high order WENO scheme in [18] for solving the static Eikonal equation as a representative example. Numerical examples are given to demonstrate the performance of these two approaches.

关键词： Fast sweeping method WENO scheme Boundary condition

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

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Communications on Applied Mathematics and Computation 2020年第3期2卷 429-460页

作者： Jun Zhu chi-wang shu College of Science Nanjing University of Aeronautics and AstronauticsNanjing 210016JiangsuChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

A new type of high-order multi-resolution weighted essentially non-oscillatory(WENO)schemes(Zhu and shu in J Comput Phys,375:659-683,2018)is applied to solve for steady-state problems on structured *** the classical WENO schemes(Jiang and shu in J Comput Phys,126:202-228,1996)might suffer from slight post-shock oscillations(which are responsible for the residue to hang at a truncation error level),this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state *** new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes,could obtain fifth-order,seventh-order,and ninth-order in smooth regions,and could gradually degrade to first-order so as to suppress spurious oscillations near strong *** linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is *** is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state *** comparison with the classical fifth-order finite-difference and finite-volume WENO schemes,the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

关键词： High-order multi-resolution WENO scheme Unequal-sized hierarchical stencil Central spatial stencil Steady-state problem

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

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Communications on Applied Mathematics and Computation 2021年第4期3卷 671-700页

作者： Zheng Sun chi-wang shu Department of Mathematics The Ohio State UniversityColumbusOH 43210USA Division of Applied Mathematics Brown UniversityProvidenceRI02912USA

A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is *** this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve *** this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral *** main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal *** propose two approaches for stabilization:the modified method and the filtering *** modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small *** linear problems,most explicit RK methods can be stabilized with either approach without accuracy ***,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong *** nonlinear problems,a filtering method is *** examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

关键词： Runge-Kutta(RK)methods Strong stability Superviscosity Hyperbolic conservation laws Discontinuous Galerkin methods

ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES Ⅱ:CELL AVERAGES AND MULTIRESOLUTION

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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