双曲和四阶方程间断有限元方法的超收敛性与误差估计
|
摘要
间断有限元方法是一类数值求解只含有一阶空间导数的双曲守恒律方程的有限元方法。局部间断有限元方法作为间断有限元方法的推广，其适用于高阶方程的求解。间断有限元方法和局部间断有限元方法是近年来提出和发展的高阶精度高分辨率的数值算法。这体现在：一方面，在解的光滑区域，得到对真解的任意高阶精度的数值逼近；另一方面，通过采用合适的限制器（如近年来提出和发展的加权本质无振荡限制器），使得在间断解（包括激波和接触间断）附近，得到陡峭且无振荡的间断过渡。
本文主要研究几类时间相关偏微分方程的间断有限元和局部间断有限元方法的超收敛性和误差估计。超收敛性能够有效保证间断有限元解与真解的误差在很长一段时间内不会增长；尤其对很密的网格，该性质体现的更为明显，表明了间断有限元方法对波的分辨能力。本文给出了一类线性四阶方程的局部间断有限元方法和非线性双曲守恒律方程的间断有限元方法的超收敛性结果，拓宽和丰富了间断有限元方法的超收敛性理论。同时，间断有限元方法的误差分析为其高精度特点提供了坚实的理论依据；所得到的多维非线性双曲守恒律方程间断有限元方法的最优先验误差估计，进一步丰富了其收敛性理论。
首先，对于一维线性四阶时间相关偏微分方程，研究了局部间断有限元方法的超收敛性。通过构造特殊的全局投影，证明了当使用交错数值流通量时，局部间断有限元解以k+3/2阶超收敛到真解的一个特殊投影，其中k≥1为有限元空间中分片多项式的次数。研究结果推广了Cheng和Shu关于局部间断有限元方法求解一维线性对流扩散方程的超收敛性工作。此外，得到了数值解、其各阶空间导数及其时间导数的最优收敛性结果。线性问题、初边值问题、非线性方程以及奇异解情形等大量的数值算例验证了方法的超收敛性和解的长时间形态。
然后，对于一维非线性时间相关双曲守恒律方程，提出和分析了间断有限元方法的超收敛性。利用Taylor展开线性化手法和数值解的先验假设，证明了当使用迎风数值流通量时，间断有限元解以k+3/2阶超收敛到真解的一个特殊投影，将线性问题间断有限元方法的超收敛性分析拓宽到了非线性问题。此外，得到了数值解及其时间导数的最优误差估计。一维多项式非线性方程、强非线性方程和二维问题等一系列数值算例验证了方法的超收敛性和解的长时间形态，表明了间断有限元方法用于处理非线性守恒律方程的计算有效性。
最后，对于多维非线性双曲守恒律方程，给出了Cartesian网格上间断有限元方法的误差分析。利用多维特殊投影的超收敛性质，证明了当使用分片k (k≥2)次张量积多项式时，间断有限元方法对于迎风数值流通量情形的最优误差估计，并且指出了对于一般单调数值流通量情形，该方法可得到次优误差估计。
总之，间断有限元方法是一类求解双曲守恒律和对流占优偏微分方程的高阶精度、高分辨率的数值方法。本文着重研究了间断有限元方法和局部间断有限元方法的超收敛性和误差估计，给出了此类方法可有效用于长时间数值模拟的坚实理论依据，进一步证实了其高阶精度特性。误差分析和数值试验显示了间断有限元方法和局部间断有限元方法用于求解线性方程、非线性方程、一维以及多维时间相关问题的计算有效性和优越性。
Discontinuous Galerkin methods are a class of finite element methods originally de-vised to numerically solve hyperbolic conservation laws containing only first order spatialderivatives. As an extension of discontinuous Galerkin methods, the local discontinuousGalerkin methods are developed with the goal of solving high order partial diferential e-quations. Discontinuous Galerkin and local discontinuous Galerkin methods are recentlyproposed and developed high order accurate high resolution numerical methods not on-ly in obtaining arbitrary high order accuracy approximation to the exact solution withinsmooth regions but also in producing sharp and non-oscillatory discontinuity transitionsnear discontinuous solutions, including shocks and contact discontinuities, by employ-ing suitable limiters (for instance, weighted essentially non-oscillatory limiters that arerecently introduced and developed).
Our attention is mainly paid to the superconvergence property and error estimatesof discontinuous Galerkin and local discontinuous Galerkin methods for solving severaltime-dependent partial diferential equations. An important motivation for investigatingsuch superconvergence is to lay a solid theoretical foundation for the fact that the errorbetween the discontinuous Galerkin solution and the exact solution does not grow over along time period. This property is especially prominent for fine meshes. The supercon-vergence property also demonstrates the capability of discontinuous Galerkin method inresolving waves. The superconvergence results presented in this work regarding the localdiscontinuous Galerkin method for a class of linear fourth-order time-dependent problemsand discontinuous Galerkin methods for nonlinear hyperbolic conservation laws broadenand reinforce the superconvergence theory of discontinuous Galerkin methods. Mean-while, error analysis of discontinuous Galerkin methods provides a solid theoretical foun-dation for its high order accuracy feature. Optimal a priori error estimates are obtainedfor discontinuous Galerkin methods applied to nonlinear hyperbolic conservation laws inmultiple dimensions, which further strengthens the convergence theory of discontinuousGalerkin methods.
We start by exploring the superconvergence property of the local discontinuousGalerkin methods for solving one-dimensional linear fourth-order time-dependent prob-lems. By constructing a special global projection, the local discontinuous Galerkin so-
lution is proved to be(k+3/2)th order superconvergent to a particular projection of theexact solution, when alternating numerical fluxes are used. Here and below, k≥1is thepiecewise polynomial degree of the finite element space. This result extends the work byCheng and Shu, in which superconvergence of the local discontinuous Galerkin methodapplied to one-dimensional linear convection-difusion equations is analyzed. Moreover,optimal convergence results on numerical solution, its spatial derivatives of diferent or-ders as well as its time derivative are obtained. Various numerical examples, includinglinear problems, initial boundary value problems, nonlinear equations and solutions hav-ing singularities, are reported to verify the superconvergence property and time evolutionof the local discontinuous Galerkin method.
We proceed to propose and analyze the superconvergence property of discontinu-ous Galerkin methods for solving one-dimensional nonlinear time-dependent hyperbolicconservation laws. By virtue of the linearized approach based on Taylor expansions andan a priori assum(ption about the numerical solution, the discontinuous Galerkin solutionis proved to be(k+3/2)
2th order superconvergent to a particular projection of the exact so-lution when upwind numerical fluxes are used. This work extends the superconvergenceanalysis of discontinuous Galerkin methods from linear problems to nonlinear ones. Inaddition, optimal error estimates on numerical solution and its time derivative are derived.A series of numerical experiments, including one-dimensional nonlinear conservationlaws with polynomial flux functions and strong nonlinearities as well as two-dimensionalcases, are presented to validate the superconvergence property and time evolution of thediscontinuous Galerkin method. The computational efciency of discontinuous Galerkinmethods for solving nonlinear conservation laws is thusly demonstrated.
We end by studying the error estimates of discontinuous Galerkin methods for multi-dimensional nonlinear hyperbolic conservation laws on Cartesian meshes. By using su-perconvergence property of multi-dimensional special projections, optimal error estimatesof discontinuous Galerkin methods for tensor product polynomials of degree at most k(k≥2) are obtained when upwind numerical fluxes are considered. It is also pointed outthat extension of the current approach to proving suboptimal error estimates for generalmonotone numerical fluxes can be easily made.
In summary, discontinuous Galerkin methods are a class of high order accurate highresolution numerical methods for solving hyperbolic conservation laws and convection-dominated partial diferential equations. The emphasis of this work is put on the study of superconvergence property and error estimates of discontinuous Galerkin and local dis-continuous Galerkin methods, which provides a solid theoretical foundation for the nicebehaviour of discontinuous Galerkin methods for long time simulations. The high orderaccuracy performance of the discontinuous Galerkin method is further confirmed. Erroranalysis together with numerical experiments shows the computational efciency and ca-pability of discontinuous Galerkin methods and local discontinuous Galerkin methods forsolving linear equations, nonlinear equations, one-and multi-dimensional time-dependentproblems.
本文主要研究几类时间相关偏微分方程的间断有限元和局部间断有限元方法的超收敛性和误差估计。超收敛性能够有效保证间断有限元解与真解的误差在很长一段时间内不会增长；尤其对很密的网格，该性质体现的更为明显，表明了间断有限元方法对波的分辨能力。本文给出了一类线性四阶方程的局部间断有限元方法和非线性双曲守恒律方程的间断有限元方法的超收敛性结果，拓宽和丰富了间断有限元方法的超收敛性理论。同时，间断有限元方法的误差分析为其高精度特点提供了坚实的理论依据；所得到的多维非线性双曲守恒律方程间断有限元方法的最优先验误差估计，进一步丰富了其收敛性理论。
首先，对于一维线性四阶时间相关偏微分方程，研究了局部间断有限元方法的超收敛性。通过构造特殊的全局投影，证明了当使用交错数值流通量时，局部间断有限元解以k+3/2阶超收敛到真解的一个特殊投影，其中k≥1为有限元空间中分片多项式的次数。研究结果推广了Cheng和Shu关于局部间断有限元方法求解一维线性对流扩散方程的超收敛性工作。此外，得到了数值解、其各阶空间导数及其时间导数的最优收敛性结果。线性问题、初边值问题、非线性方程以及奇异解情形等大量的数值算例验证了方法的超收敛性和解的长时间形态。
然后，对于一维非线性时间相关双曲守恒律方程，提出和分析了间断有限元方法的超收敛性。利用Taylor展开线性化手法和数值解的先验假设，证明了当使用迎风数值流通量时，间断有限元解以k+3/2阶超收敛到真解的一个特殊投影，将线性问题间断有限元方法的超收敛性分析拓宽到了非线性问题。此外，得到了数值解及其时间导数的最优误差估计。一维多项式非线性方程、强非线性方程和二维问题等一系列数值算例验证了方法的超收敛性和解的长时间形态，表明了间断有限元方法用于处理非线性守恒律方程的计算有效性。
最后，对于多维非线性双曲守恒律方程，给出了Cartesian网格上间断有限元方法的误差分析。利用多维特殊投影的超收敛性质，证明了当使用分片k (k≥2)次张量积多项式时，间断有限元方法对于迎风数值流通量情形的最优误差估计，并且指出了对于一般单调数值流通量情形，该方法可得到次优误差估计。
总之，间断有限元方法是一类求解双曲守恒律和对流占优偏微分方程的高阶精度、高分辨率的数值方法。本文着重研究了间断有限元方法和局部间断有限元方法的超收敛性和误差估计，给出了此类方法可有效用于长时间数值模拟的坚实理论依据，进一步证实了其高阶精度特性。误差分析和数值试验显示了间断有限元方法和局部间断有限元方法用于求解线性方程、非线性方程、一维以及多维时间相关问题的计算有效性和优越性。
Discontinuous Galerkin methods are a class of finite element methods originally de-vised to numerically solve hyperbolic conservation laws containing only first order spatialderivatives. As an extension of discontinuous Galerkin methods, the local discontinuousGalerkin methods are developed with the goal of solving high order partial diferential e-quations. Discontinuous Galerkin and local discontinuous Galerkin methods are recentlyproposed and developed high order accurate high resolution numerical methods not on-ly in obtaining arbitrary high order accuracy approximation to the exact solution withinsmooth regions but also in producing sharp and non-oscillatory discontinuity transitionsnear discontinuous solutions, including shocks and contact discontinuities, by employ-ing suitable limiters (for instance, weighted essentially non-oscillatory limiters that arerecently introduced and developed).
Our attention is mainly paid to the superconvergence property and error estimatesof discontinuous Galerkin and local discontinuous Galerkin methods for solving severaltime-dependent partial diferential equations. An important motivation for investigatingsuch superconvergence is to lay a solid theoretical foundation for the fact that the errorbetween the discontinuous Galerkin solution and the exact solution does not grow over along time period. This property is especially prominent for fine meshes. The supercon-vergence property also demonstrates the capability of discontinuous Galerkin method inresolving waves. The superconvergence results presented in this work regarding the localdiscontinuous Galerkin method for a class of linear fourth-order time-dependent problemsand discontinuous Galerkin methods for nonlinear hyperbolic conservation laws broadenand reinforce the superconvergence theory of discontinuous Galerkin methods. Mean-while, error analysis of discontinuous Galerkin methods provides a solid theoretical foun-dation for its high order accuracy feature. Optimal a priori error estimates are obtainedfor discontinuous Galerkin methods applied to nonlinear hyperbolic conservation laws inmultiple dimensions, which further strengthens the convergence theory of discontinuousGalerkin methods.
We start by exploring the superconvergence property of the local discontinuousGalerkin methods for solving one-dimensional linear fourth-order time-dependent prob-lems. By constructing a special global projection, the local discontinuous Galerkin so-
lution is proved to be(k+3/2)th order superconvergent to a particular projection of theexact solution, when alternating numerical fluxes are used. Here and below, k≥1is thepiecewise polynomial degree of the finite element space. This result extends the work byCheng and Shu, in which superconvergence of the local discontinuous Galerkin methodapplied to one-dimensional linear convection-difusion equations is analyzed. Moreover,optimal convergence results on numerical solution, its spatial derivatives of diferent or-ders as well as its time derivative are obtained. Various numerical examples, includinglinear problems, initial boundary value problems, nonlinear equations and solutions hav-ing singularities, are reported to verify the superconvergence property and time evolutionof the local discontinuous Galerkin method.
We proceed to propose and analyze the superconvergence property of discontinu-ous Galerkin methods for solving one-dimensional nonlinear time-dependent hyperbolicconservation laws. By virtue of the linearized approach based on Taylor expansions andan a priori assum(ption about the numerical solution, the discontinuous Galerkin solutionis proved to be(k+3/2)
2th order superconvergent to a particular projection of the exact so-lution when upwind numerical fluxes are used. This work extends the superconvergenceanalysis of discontinuous Galerkin methods from linear problems to nonlinear ones. Inaddition, optimal error estimates on numerical solution and its time derivative are derived.A series of numerical experiments, including one-dimensional nonlinear conservationlaws with polynomial flux functions and strong nonlinearities as well as two-dimensionalcases, are presented to validate the superconvergence property and time evolution of thediscontinuous Galerkin method. The computational efciency of discontinuous Galerkinmethods for solving nonlinear conservation laws is thusly demonstrated.
We end by studying the error estimates of discontinuous Galerkin methods for multi-dimensional nonlinear hyperbolic conservation laws on Cartesian meshes. By using su-perconvergence property of multi-dimensional special projections, optimal error estimatesof discontinuous Galerkin methods for tensor product polynomials of degree at most k(k≥2) are obtained when upwind numerical fluxes are considered. It is also pointed outthat extension of the current approach to proving suboptimal error estimates for generalmonotone numerical fluxes can be easily made.
In summary, discontinuous Galerkin methods are a class of high order accurate highresolution numerical methods for solving hyperbolic conservation laws and convection-dominated partial diferential equations. The emphasis of this work is put on the study of superconvergence property and error estimates of discontinuous Galerkin and local dis-continuous Galerkin methods, which provides a solid theoretical foundation for the nicebehaviour of discontinuous Galerkin methods for long time simulations. The high orderaccuracy performance of the discontinuous Galerkin method is further confirmed. Erroranalysis together with numerical experiments shows the computational efciency and ca-pability of discontinuous Galerkin methods and local discontinuous Galerkin methods forsolving linear equations, nonlinear equations, one-and multi-dimensional time-dependentproblems.
引文
[1] Lax P D. Weak Solutions of Nonlinear Hyperbolic Equations and Their Numer-ical Computation[J]. Communications on Pure and Applied Mathematics,1954,7(1):159–193.
[2] Lax P D. Hyperbolic Systems of Conservation Laws II[J]. Communications onPure and Applied Mathematics,1957,10(4):537–566.
[3] Morawetz C S. A Weak Solution for a System of Equations of Elliptic-HyperbolicType[J]. Communications on Pure and Applied Mathematics,1958,11(3):315–331.
[4] Lax P D. Hyperbolic Systems of Conservation Laws and the Mathematical Theoryof Shock Waves[M].1st. Philadelphia: SIAM,1972:1–54.
[5] Harten A, Hyman J M, Lax P D, et al. On Finite-Diference Approximations andEntropy Conditions for Shocks[J]. Communications on Pure and Applied Mathe-matics,1976,29(3):297–322.
[6] Tadmor E. Numerical Viscosity and the Entropy Condition for Conservative Dif-ference Schemes[J]. Mathematics of Computation,1984,43(168):369–381.
[7] Godunov S K. A Diference Method for Numerical Calculation of DiscontinuousSolutions of the Equations of Hydrodynamics[J]. Matematicheskii Sbornik,1959,89(3):271–306.
[8] Friedrichs K O, Lax P D. Systems of Conservation Equations with a Convex Ex-tension[J]. Proceedings of the National Academy of Sciences,1971,68(8):1686–1688.
[9] Roe P L. Approximate Riemann Solvers, Parameter Vectors, and DiferenceSchemes[J]. Journal of Computational Physics,1981,43(2):357–372.
[10] Engquist B, Osher S. One-Sided Diference Approximations for Nonlinear Con-servation Laws[J]. Mathematics of Computation,1981,36(154):321–351.
[11] Harten A. High Resolution Schemes for Hyperbolic Conservation Laws[J]. Journalof Computational Physics,1983,49(3):357–393.
[12] Shu C W. TVB Uniformly High-Order Schemes for Conservation Laws[J]. Math-ematics of Computation,1987,49(179):105–121.
[13] Harten A, Engquist B, Osher S, et al. Uniformly High Order Accurate Essen-tially Non-Oscillatory Schemes, III[J]. Journal of Computational Physics,1987,71(2):231–303.
[14] Cockburn B, Shu C W. TVB Runge-Kutta Local Projection Discontinuous GalerkinFinite Element Method for Conservation Laws II: General Framework[J]. Mathe-matics of Computation,1989,52(186):411–435.
[15] Liu X, Osher S, Chan T. Weighted Essentially Non-Oscillatory Schemes[J]. Jour-nal of Computational Physics,1994,115(1):200–212.
[16] Shu C W. High Order Weighted Essentially Nonoscillatory Schemes for Convec-tion Dominated Problems[J]. SIAM Review,2009,51(1):82–126.
[17] Shu C W. On High-Order Accurate Weighted Essentially Non-Oscillatory and Dis-continuous Galerkin Schemes for Compressible Turbulence Simulations[J]. Philo-sophical Transactions of the Royal Society A: Mathematical, Physical and Engi-neering Sciences,2013,371(1982).
[18] Reed W H, Hill T R. Triangular Mesh Methods for the Neutron Transport Equa-tion[J]. Los Alamos Report LA-UR-73-479,1973.
[19] Cockburn B, Shu C W. The Runge-Kutta Local Projection P1-DiscontinuousGalerkin Finite Element Method for Scalar Conservation Laws[J]. MathematicalModelling and Numerical Analysis,1991,25(3):337–361.
[20] Cockburn B, Lin S Y, Shu C W. TVB Runge-Kutta Local Projection Discontinu-ous Galerkin Finite Element Method for Conservation Laws III: One-DimensionalSystems[J]. Journal of Computational Physics,1989,84(1):90–113.
[21] Cockburn B, Hou S, Shu C W. The Runge-Kutta Local Projection DiscontinuousGalerkin Finite Element Method for Conservation Laws IV: The MultidimensionalCase[J]. Mathematics of Computation,1990,54(190):545–581.
[22] Cockburn B, Shu C W. The Runge–Kutta Discontinuous Galerkin Method forConservation Laws V: Multidimensional Systems[J]. Journal of ComputationalPhysics,1998,141(2):199–224.
[23] Shu C W, Osher S. Efcient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes[J]. Journal of Computational Physics,1988,77(2):439–471.
[24] Qiu J, Shu C W. Runge–Kutta Discontinuous Galerkin Method Using WENOLimiters[J]. SIAM Journal on Scientific Computing,2005,26(3):907–929.
[25] Zhu J, Qiu J, Shu C W, et al. Runge–Kutta Discontinuous Galerkin Method UsingWENO Limiters II: Unstructured Meshes[J]. Journal of Computational Physics,2008,227(9):4330–4353.
[26] Zhu J, Qiu J. Local DG Method Using WENO Type Limiters for Convection-Difusion Problems[J]. Journal of Computational Physics,2011,230(11):4353–4375.
[27] Zhu J, Liu T, Qiu J, et al. RKDG Methods with WENO Limiters for UnsteadyCavitating Flow[J]. Computers&Fluids,2012,57:52–65.
[28] Zhao J, Tang H Z. Runge-Kutta Discontinuous Galerkin Methods with WENOLimiter for the Special Relativistic Hydrodynamics[J]. Journal of ComputationalPhysics,2013,242:138–168.
[29] Qiu J, Shu C W. Hermite WENO Schemes and Their Application as Limiters forRunge–Kutta Discontinuous Galerkin Method: One-Dimensional Case[J]. Journalof Computational Physics,2004,193(1):115–135.
[30] Zhu J, Qiu J. Hermite WENO Schemes and Their Application as Limiters forRunge-Kutta Discontinuous Galerkin Method, III: Unstructured Meshes[J]. Jour-nal of Scientific Computing,2009,39(2):293–321.
[31] Jiang Z H, Yan C, Yu J, et al. Hermite WENO-Based Limiters for High OrderDiscontinuous Galerkin Method on Unstructured Grids[J]. Acta Mechanica Sinica,2012,28(2):241–252.
[32] Luo H, Xia Y, Li S, et al. A Hermite WENO Reconstruction-Based Discontinu-ous Galerkin Method for the Euler Equations on Tetrahedral Grids[J]. Journal ofComputational Physics,2012,231(16):5489–5503.
[33] Zhong X, Shu C W. A Simple Weighted Essentially Nonoscillatory Limiterfor Runge-Kutta Discontinuous Galerkin Methods[J]. Journal of ComputationalPhysics,2013,232(1):397–415.
[34] Zhu J, Zhong X, Shu C W, et al. Runge-Kutta Discontinuous Galerkin MethodUsing a New Type of WENO Limiters on Unstructured Meshes[J]. Journal ofComputational Physics,2013,248:200–220.
[35] Zhang X, Shu C W. On Maximum-Principle-Satisfying High Order Schemesfor Scalar Conservation Laws[J]. Journal of Computational Physics,2010,229(9):3091–3120.
[36] Zhang X, Shu C W. On Positivity-Preserving High Order Discontinuous GalerkinSchemes for Compressible Euler Equations on Rectangular Meshes[J]. Journal ofComputational Physics,2010,229(23):8918–8934.
[37] Zhang X, Shu C W. Positivity-Preserving High Order Discontinuous GalerkinSchemes for Compressible Euler Equations with Source Terms[J]. Journal of Com-putational Physics,2011,230(4):1238–1248.
[38] Zhang X, Xia Y, Shu C W. Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Lawson Triangular Meshes[J]. Journal of Scientific Computing,2012,50(1):29–62.
[39] Zhang Y, Zhang X, Shu C W. Maximum-Principle-Satisfying Second Order Dis-continuous Galerkin Schemes for Convection-Difusion Equations on TriangularMeshes[J]. Journal of Computational Physics,2013,234:295–316.
[40] Xing Y, Zhang X, Shu C W. Positivity-Preserving High Order Well-Balanced Dis-continuous Galerkin Methods for the Shallow Water Equations[J]. Advances inWater Resources,2010,33(12):1476–1493.
[41] Qiu J M, Shu C W. Positivity Preserving Semi-Lagrangian Discontinuous GalerkinFormulation: Theoretical Analysis and Application to the Vlasov–Poisson Sys-tem[J]. Journal of Computational Physics,2011,230(23):8386–8409.
[42] Cheng Y, Li F, Qiu J, et al. Positivity-Preserving DG and Central DG Methods forIdeal MHD Equations[J]. Journal of Computational Physics,2013,238:255–280.
[43] Bassi F, Rebay S. A High-Order Accurate Discontinuous Finite Element Methodfor the Numerical Solution of the Compressible Navier–Stokes Equations[J]. Jour-nal of Computational Physics,1997,131(2):267–279.
[44] Cockburn B, Shu C W. The Local Discontinuous Galerkin Method for Time-Dependent Convection-Difusion Systems[J]. SIAM Journal on Numerical Analy-sis,1998,35(6):2440–2463.
[45] Yan J, Shu C W. A Local Discontinuous Galerkin Method for KdV Type Equation-s[J]. SIAM Journal on Numerical Analysis,2002,40(2):769–791.
[46] Yan J, Shu C W. Local Discontinuous Galerkin Methods for Partial Diferential E-quations with Higher Order Derivatives[J]. Journal of Scientific Computing,2002,17(1):27–47.
[47] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for Two Classes of Two-Dimensional Nonlinear Wave Equations[J]. Physica D,2005,208(1):21–58.
[48] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for the Kuramoto–Sivashinsky Equations and the Ito-Type Coupled KdV Equations[J]. ComputerMethods in Applied Mechanics and Engineering,2006,195(25):3430–3447.
[49] Cockburn B, Shu C W. Runge–Kutta Discontinuous Galerkin Methods forConvection-Dominated Problems[J]. Journal of Scientific Computing,2001,16(3):173–261.
[50] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Diferential Equations[J]. Communications in ComputationalPhysics,2010,7(1):1–46.
[51] Cheng Y, Shu C W. A Discontinuous Galerkin Finite Element Method for TimeDependent Partial Diferential Equations with Higher Order Derivatives[J]. Math-ematics of Computation,2008,77(262):699–730.
[52] Castillo P, Cockburn B, Perugia I, et al. An a Priori Error Analysis of the Local Dis-continuous Galerkin Method for Elliptic Problems[J]. SIAM Journal on NumericalAnalysis,2000,38(5):1676–1706.
[53] Cockburn B, Gopalakrishnan J, Lazarov R. Unified Hybridization of Discontinu-ous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order EllipticProblems[J]. SIAM Journal on Numerical Analysis,2009,47(2):1319–1365.
[54] Wihler T P, Rivie`re B. Discontinuous Galerkin Methods for Second-Order EllipticPDE with Low-Regularity Solutions[J]. Journal of Scientific Computing,2011,46(2):151–165.
[55] Babusˇka I. The Finite Element Method with Penalty[J]. Mathematics of Compu-tation,1973,27(122):221–228.
[56] Babusˇka I, Zla′mal M. Nonconforming Elements in the Finite Element Methodwith Penalty[J]. SIAM Journal on Numerical Analysis,1973,10(5):863–875.
[57] Baker G A. Finite Element Methods for Elliptic Equations Using NonconformingElements[J]. Mathematics of Computation,1977,31(137):45–59.
[58] Douglas J, Dupont T. Interior Penalty Procedures for Elliptic and ParabolicGalerkin Methods[J]. Computing Methods in Applied Sciences,1976,58:207–216.
[59] Arnold D. An Interior Penalty Finite Element Method with Discontinuous Ele-ments[J]. SIAM Journal on Numerical Analysis,1982,19(4):742–760.
[60] Arnold D, Brezzi F, Cockburn B, et al. Unified Analysis of Discontinuous GalerkinMethods for Elliptic Problems[J]. SIAM Journal on Numerical Analysis,2002,39(5):1749–1779.
[61] Oden J T, Babusˇka I, Baumann C E. A Discontinuous hp Finite Element Methodfor Difusion Problems[J]. Journal of Computational Physics,1998,146(2):491–519.
[62] Gudi T. A New Error Analysis for Discontinuous Finite Element Methods forLinear Elliptic Problems[J]. Mathematics of Computation,2010,79(272):2169–2189.
[63] Rivie`re B. Discontinuous Galerkin Methods for Solving Elliptic and ParabolicEquations: Theory and Implementation[M].1st. Philadelphia: SIAM,2008:1–157.
[64] LeSaint P, Raviart P A. On a Finite Element Method for Solving the NeutronTransport Equation[M].1st. New York: Academic Press,1974:89–145.
[65] Johnson C, Pitka¨ranta J. An Analysis of the Discontinuous Galerkin Method fora Scalar Hyperbolic Equation.[J]. Mathematics of Computation,1986,46(173):1–26.
[66] Peterson T. A Note on the Convergence of the Discontinuous Galerkin Methodfor a Scalar Hyperbolic Equation[J]. SIAM Journal on Numerical Analysis,1991,28(1):133–140.
[67] Richter G R. An Optimal-Order Error Estimate for the Discontinuous GalerkinMethod.[J]. Mathematics of Computation,1988,50(181):75–88.
[68] Cockburn B, Dong B, Guzma′n J. Optimal Convergence of the Original DG Methodfor the Transport-Reaction Equation on Special Meshes[J]. SIAM Journal on Nu-merical Analysis,2008,46(3):1250–1265.
[69] Cockburn B, Dong B, Guzma′n J, et al. Optimal Convergence of the Original DGMethod on Special Meshes for Variable Transport Velocity[J]. SIAM Journal onNumerical Analysis,2010,48(1):133–146.
[70] Zhang Q, Shu C W. Stability Analysis and a Priori Error Estimates of the Third Or-der Explicit Runge-Kutta Discontinuous Galerkin Method for Scalar ConservationLaws[J]. SIAM Journal on Numerical Analysis,2010,48(3):1038–1063.
[71] Zhang Q, Shu C W. Error Estimates to Smooth Solutions of Runge–Kutta Dis-continuous Galerkin Methods for Scalar Conservation Laws[J]. SIAM Journal onNumerical Analysis,2004,42(2):641–666.
[72] Zhang Q, Shu C W. Error Estimates to Smooth Solutions of Runge-Kutta Dis-continuous Galerkin Method for Symmetrizable Systems of Conservation Laws[J].SIAM Journal on Numerical Analysis,2006,44(4):1703–1720.
[73] Cockburn B, Guzma′n J. Error Estimates for the Runge-Kutta DiscontinuousGalerkin Method for the Transport Equation with Discontinuous Initial Data[J].SIAM Journal on Numerical Analysis,2008,46(3):1364–1398.
[74] Zhang Q, Shu C W. Error Estimates for the Third Order Explicit Runge-Kutta Dis-continuous Galerkin Method for Linear Hyperbolic Equation in One-Dimensionwith Discontinuous Initial Data[J]. Numerische Mathematik, In press,2013.
[75] Yang Y, Shu C W. Discontinuous Galerkin Method for Hyperbolic Equations In-volving Delta-Singularities: Negative-Order Norm Error Estimates and Applica-tions[J]. Numerische Mathematik,2013,124(4):753–781.
[76] Jiang G, Shu C W. On a Cell Entropy Inequality for Discontinuous Galerkin Meth-ods[J]. Mathematics of Computation,1994,62(206):531–538.
[77] Cockburn B, Gremaud P A. Error Estimates for Finite Element Methods for ScalarConservation Laws[J]. SIAM Journal on Numerical Analysis,1996,33(2):522–554.
[78] Baccouch M, Adjerid S. Discontinuous Galerkin Error Estimation for HyperbolicProblems on Unstructured Triangular Meshes[J]. Computer Methods in AppliedMechanics and Engineering,2011,200(1):162–177.
[79] Xu Y, Shu C W. Error Estimates of the Semi-Discrete Local Discontinu-ous Galerkin Method for Nonlinear Convection-Difusion and KdV Equations[J].Computer Methods in Applied Mechanics and Engineering,2007,196(37):3805–3822.
[80] Xu Y, Shu C W. Optimal Error Estimates of the Semidiscrete Local DiscontinuousGalerkin Methods for High Order Wave Equations[J]. SIAM Journal on NumericalAnalysis,2012,50(1):79–104.
[81] Cockburn B, Dong B. An Analysis of the Minimal Dissipation Local Discontinu-ous Galerkin Method for Convection-Difusion Problems[J]. Journal of ScientificComputing,2007,32(2):233–262.
[82] Feistauer M, Kucˇera V, Najzar K, et al. Analysis of Space-Time DiscontinuousGalerkin Method for Nonlinear Convection-Difusion Problems[J]. NumerischeMathematik,2011,117(2):251–288.
[83] Dong B, Shu C W. Analysis of a Local Discontinuous Galerkin Method for LinearTime-Dependent Fourth-Order Problems[J]. SIAM Journal on Numerical Analy-sis,2009,47(5):3240–3268.
[84] Bona J L, Chen H, Karakashian O, et al. Conservative Discontinuous GalerkinMethods for the Generalized Korteweg–de Vries Equation[J]. Mathematics ofComputation,2013,82(283):1401–1432.
[85] Adjerid S, Devine K, Flaherty J, et al. A Posteriori Error Estimation for Discontin-uous Galerkin Solutions of Hyperbolic Problems[J]. Computer Methods in AppliedMechanics and Engineering,2002,191(11):1097–1112.
[86] Adjerid S, Massey T. Superconvergence of Discontinuous Galerkin Solutions for aNonlinear Scalar Hyperbolic Problem[J]. Computer Methods in Applied Mechan-ics and Engineering,2006,195(25):3331–3346.
[87] Adjerid S, Klauser A. Superconvergence of Discontinuous Finite Element Solu-tions for Transient Convection-Difusion Problems[J]. Journal of Scientific Com-puting,2005,22(1):5–24.
[88] Huang Q, Xie H, Brunner H. Superconvergence of Discontinuous Galerkin Solu-tions for Delay Diferential Equations of Pantograph Type[J]. SIAM Journal onScientific Computing,2011,33(5):2664–2684.
[89] Celiker F, Cockburn B. Superconvergence of the Numerical Traces of Discontinu-ous Galerkin and Hybridized Methods for Convection-Difusion Problems in OneSpace Dimension[J]. Mathematics of Computation,2007,76(257):67–96.
[90] Cheng Y, Shu C W. Superconvergence and Time Evolution of DiscontinuousGalerkin Finite Element Solutions[J]. Journal of Computational Physics,2008,227(22):9612–9627.
[91] Cheng Y, Shu C W. Superconvergence of Local Discontinuous Galerkin Methodsfor One-Dimensional Convection-Difusion Equations[J]. Computers&Structures,2009,87(11):630–641.
[92] Zhong X, Shu C W. Numerical Resolution of Discontinuous Galerkin Methods forTime Dependent Wave Equations[J]. Computer Methods in Applied Mechanicsand Engineering,2011,200(41):2814–2827.
[93] Cheng Y, Shu C W. Superconvergence of Discontinuous Galerkin and Local Dis-continuous Galerkin Schemes for Linear Hyperbolic and Convection-Difusion E-quations in One Space Dimension[J]. SIAM Journal on Numerical Analysis,2010,47(6):4044–4072.
[94] Huford C, Xing Y. Superconvergence of the Local Discontinuous Galerkin Methodfor the Linearized Korteweg–de Vries Equation[J]. Journal of Computational andApplied Mathematics,2014,255:441–455.
[95] Guo W, Zhong X, Qiu J M. Superconvergence of Discontinuous Galerkin and Lo-cal Discontinuous Galerkin Methods: Eigen-Structure Analysis Based on FourierApproach[J]. Journal of Computational Physics,2013,235:458–485.
[96] Ketcheson D I, Macdonald C B, Gottlieb S. Optimal Implicit Strong Stabili-ty Preserving Runge-Kutta Methods[J]. Applied Numerical Mathematics,2009,59(2):373–392.
[97] Gottlieb S, Ketcheson D I, Shu C W. Strong Stability Preserving Runge-Kutta andMultistep Time Discretizations[M].1st. Singapore: World Scientific,2011:1–164.
[98] Ketcheson D I, Gottlieb S, Macdonald C B. Strong Stability Preserving Two-Step Runge-Kutta Methods[J]. SIAM Journal on Numerical Analysis,2011,49(6):2618–2639.
[99] Xia Y, Xu Y, Shu C W. Efcient Time Discretization for Local DiscontinuousGalerkin Methods[J]. Discrete and Continuous Dynamical Systems–Series B,2007,8(3):677–693.
[100] Christlieb A, Ong B, Qiu J M. Integral Deferred Correction Methods Constructedwith High Order Runge-Kutta Integrators[J]. Mathematics of Computation,2010,79(270):761–783.
[101] Christlieb A, Morton M, Ong B, et al. Semi-Implicit Integral Deferred CorrectionConstructed with Additive Runge-Kutta Methods[J]. Communications in Mathe-matical Sciences,2011,9(3):879–902.
[102] Christlieb A J, Haynes R D, Ong B W. A Parallel Space-Time Algorithm[J]. SIAMJournal on Scientific Computing,2012,34(5):C233–C248.
[103] Guo R, Xu Y. Efcient Solvers of Discontinuous Galerkin Discretization for theCahn–Hilliard Equations[J]. Journal of Scientific Computing, In press,2013.
[104] Su¨li E, Mozolevski I. hp-Version Interior Penalty DGFEMs for the BiharmonicEquation[J]. Computer Methods in Applied Mechanics and Engineering,2007,196(13):1851–1863.
[105] Elliott C M, Zheng S. On the Cahn–Hilliard Equation[J]. Archive for RationalMechanics and Analysis,1986,96(4):339–357.
[106] Feng X, Prohl A. Error Analysis of a Mixed Finite Element Method for the Cahn–Hilliard Equation[J]. Numerische Mathematik,2004,99(1):47–84.
[107] Zhang S, Wang M. A Nonconforming Finite Element Method for the Cahn–Hilliard Equation[J]. Journal of Computational Physics,2010,229(19):7361–7372.
[108] Brenner S C, Gu S, Gudi T, et al. A Quadratic C0Interior Penalty Method forLinear Fourth Order Boundary Value Problems with Boundary Conditions of theCahn–Hilliard Type[J]. SIAM Journal on Numerical Analysis,2012,50(4):2088–2110.
[109] Georgoulis E H, Houston P, Virtanen J. An a Posteriori Error Indicator for Dis-continuous Galerkin Approximations of Fourth-Order Elliptic Problems[J]. IMAJournal of Numerical Analysis,2011,31(1):281–298.
[110] Brenner S, Gudi T, Sung L. An a Posteriori Error Estimator for a Quadratic C0-Interior Penalty Method for the Biharmonic Problem[J]. IMA Journal of NumericalAnalysis,2010,30(3):777–798.
[111] Cockburn B, Dong B, Guzma′n J. A Hybridizable and Superconvergent Discontin-uous Galerkin Method for Biharmonic Problems[J]. Journal of Scientific Comput-ing,2009,40(1):141–187.
[112] Castillo P, Cockburn B, Scho¨tzau D, et al. Optimal a Priori Error Estimates for thehp-Version of the Local Discontinuous Galerkin Method for Convection-DifusionProblems[J]. Mathematics of Computation,2002,71(238):455–478.
[113] Ciarlet P G. The Finite Element Method for Elliptic Problems[M].1st. Amsterdam:North Holland,1978:110–332.
[114] Cockburn B, Luskin M, Shu C W, et al. Enhanced Accuracy by Post-Processingfor Finite Element Methods for Hyperbolic Equations[J]. Mathematics of Compu-tation,2003,72(242):577–606.
[115] Walfisch D, Ryan J K, Kirby R M, et al. One-Sided Smoothness-IncreasingAccuracy-Conserving Filtering for Enhanced Streamline Integration Through Dis-continuous Fields[J]. Journal of Scientific Computing,2009,38(2):164–184.
[116] van Slingerland P, Ryan J K, Vuik C. Position-Dependent Smoothness-IncreasingAccuracy-Conserving (SIAC) Filtering for Improving Discontinuous Galerkin So-lutions[J]. SIAM Journal on Scientific Computing,2011,33(2):802–825.
[117] Mirzaee H, Ryan J K, Kirby R M. Efcient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin So-lutions[J]. Journal of Scientific Computing,2012,52(1):85–112.
[118] King J, Mirzaee H, Ryan J K, et al. Smoothness-Increasing Accuracy-Conserving(SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors versusHigher-Order Accuracy[J]. Journal of Scientific Computing,2012,53(1):129–149.
[119] Mirzaee H, Ji L, Ryan J K, et al. Smoothness-Increasing Accuracy-Conserving(SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Tri-angular Meshes[J]. SIAM Journal on Numerical Analysis,2011,49(5):1899–1920.
[120] Ji L, Xu Y, Ryan J K. Negative-Order Norm Estimates for Nonlinear HyperbolicConservation Laws[J]. Journal of Scientific Computing,2013,54(2-3):531–548.
[121] Ji L, Xu Y, Ryan J K. Accuracy-Enhancement of Discontinuous Galerkin Solutionsfor Convection-Difusion Equations in Multiple-Dimensions[J]. Mathematics ofComputation,2012,81(280):1929–1950.
[122] LeVeque R. Numerical Methods for Conservation Laws[M].2nd. Basel:Birkha¨user Verlag,2005:97–207.
[123] Brenner S, Scott L R. The Mathematical Theory of Finite Element Methods[M].3rd. New York: Springer,2007:105–110.
[124] Gottlieb S, Ketcheson D I, Shu C W. High Order Strong Stability Preserving TimeDiscretizations[J]. Journal of Scientific Computing,2009,38(3):251–289.
[125] Cockburn B, Kanschat G, Perugia I, et al. Superconvergence of the Local Dis-continuous Galerkin Method for Elliptic Problems on Cartesian Grids[J]. SIAMJournal on Numerical Analysis,2001,39(1):264–285.
[2] Lax P D. Hyperbolic Systems of Conservation Laws II[J]. Communications onPure and Applied Mathematics,1957,10(4):537–566.
[3] Morawetz C S. A Weak Solution for a System of Equations of Elliptic-HyperbolicType[J]. Communications on Pure and Applied Mathematics,1958,11(3):315–331.
[4] Lax P D. Hyperbolic Systems of Conservation Laws and the Mathematical Theoryof Shock Waves[M].1st. Philadelphia: SIAM,1972:1–54.
[5] Harten A, Hyman J M, Lax P D, et al. On Finite-Diference Approximations andEntropy Conditions for Shocks[J]. Communications on Pure and Applied Mathe-matics,1976,29(3):297–322.
[6] Tadmor E. Numerical Viscosity and the Entropy Condition for Conservative Dif-ference Schemes[J]. Mathematics of Computation,1984,43(168):369–381.
[7] Godunov S K. A Diference Method for Numerical Calculation of DiscontinuousSolutions of the Equations of Hydrodynamics[J]. Matematicheskii Sbornik,1959,89(3):271–306.
[8] Friedrichs K O, Lax P D. Systems of Conservation Equations with a Convex Ex-tension[J]. Proceedings of the National Academy of Sciences,1971,68(8):1686–1688.
[9] Roe P L. Approximate Riemann Solvers, Parameter Vectors, and DiferenceSchemes[J]. Journal of Computational Physics,1981,43(2):357–372.
[10] Engquist B, Osher S. One-Sided Diference Approximations for Nonlinear Con-servation Laws[J]. Mathematics of Computation,1981,36(154):321–351.
[11] Harten A. High Resolution Schemes for Hyperbolic Conservation Laws[J]. Journalof Computational Physics,1983,49(3):357–393.
[12] Shu C W. TVB Uniformly High-Order Schemes for Conservation Laws[J]. Math-ematics of Computation,1987,49(179):105–121.
[13] Harten A, Engquist B, Osher S, et al. Uniformly High Order Accurate Essen-tially Non-Oscillatory Schemes, III[J]. Journal of Computational Physics,1987,71(2):231–303.
[14] Cockburn B, Shu C W. TVB Runge-Kutta Local Projection Discontinuous GalerkinFinite Element Method for Conservation Laws II: General Framework[J]. Mathe-matics of Computation,1989,52(186):411–435.
[15] Liu X, Osher S, Chan T. Weighted Essentially Non-Oscillatory Schemes[J]. Jour-nal of Computational Physics,1994,115(1):200–212.
[16] Shu C W. High Order Weighted Essentially Nonoscillatory Schemes for Convec-tion Dominated Problems[J]. SIAM Review,2009,51(1):82–126.
[17] Shu C W. On High-Order Accurate Weighted Essentially Non-Oscillatory and Dis-continuous Galerkin Schemes for Compressible Turbulence Simulations[J]. Philo-sophical Transactions of the Royal Society A: Mathematical, Physical and Engi-neering Sciences,2013,371(1982).
[18] Reed W H, Hill T R. Triangular Mesh Methods for the Neutron Transport Equa-tion[J]. Los Alamos Report LA-UR-73-479,1973.
[19] Cockburn B, Shu C W. The Runge-Kutta Local Projection P1-DiscontinuousGalerkin Finite Element Method for Scalar Conservation Laws[J]. MathematicalModelling and Numerical Analysis,1991,25(3):337–361.
[20] Cockburn B, Lin S Y, Shu C W. TVB Runge-Kutta Local Projection Discontinu-ous Galerkin Finite Element Method for Conservation Laws III: One-DimensionalSystems[J]. Journal of Computational Physics,1989,84(1):90–113.
[21] Cockburn B, Hou S, Shu C W. The Runge-Kutta Local Projection DiscontinuousGalerkin Finite Element Method for Conservation Laws IV: The MultidimensionalCase[J]. Mathematics of Computation,1990,54(190):545–581.
[22] Cockburn B, Shu C W. The Runge–Kutta Discontinuous Galerkin Method forConservation Laws V: Multidimensional Systems[J]. Journal of ComputationalPhysics,1998,141(2):199–224.
[23] Shu C W, Osher S. Efcient Implementation of Essentially Non-Oscillatory Shock-Capturing Schemes[J]. Journal of Computational Physics,1988,77(2):439–471.
[24] Qiu J, Shu C W. Runge–Kutta Discontinuous Galerkin Method Using WENOLimiters[J]. SIAM Journal on Scientific Computing,2005,26(3):907–929.
[25] Zhu J, Qiu J, Shu C W, et al. Runge–Kutta Discontinuous Galerkin Method UsingWENO Limiters II: Unstructured Meshes[J]. Journal of Computational Physics,2008,227(9):4330–4353.
[26] Zhu J, Qiu J. Local DG Method Using WENO Type Limiters for Convection-Difusion Problems[J]. Journal of Computational Physics,2011,230(11):4353–4375.
[27] Zhu J, Liu T, Qiu J, et al. RKDG Methods with WENO Limiters for UnsteadyCavitating Flow[J]. Computers&Fluids,2012,57:52–65.
[28] Zhao J, Tang H Z. Runge-Kutta Discontinuous Galerkin Methods with WENOLimiter for the Special Relativistic Hydrodynamics[J]. Journal of ComputationalPhysics,2013,242:138–168.
[29] Qiu J, Shu C W. Hermite WENO Schemes and Their Application as Limiters forRunge–Kutta Discontinuous Galerkin Method: One-Dimensional Case[J]. Journalof Computational Physics,2004,193(1):115–135.
[30] Zhu J, Qiu J. Hermite WENO Schemes and Their Application as Limiters forRunge-Kutta Discontinuous Galerkin Method, III: Unstructured Meshes[J]. Jour-nal of Scientific Computing,2009,39(2):293–321.
[31] Jiang Z H, Yan C, Yu J, et al. Hermite WENO-Based Limiters for High OrderDiscontinuous Galerkin Method on Unstructured Grids[J]. Acta Mechanica Sinica,2012,28(2):241–252.
[32] Luo H, Xia Y, Li S, et al. A Hermite WENO Reconstruction-Based Discontinu-ous Galerkin Method for the Euler Equations on Tetrahedral Grids[J]. Journal ofComputational Physics,2012,231(16):5489–5503.
[33] Zhong X, Shu C W. A Simple Weighted Essentially Nonoscillatory Limiterfor Runge-Kutta Discontinuous Galerkin Methods[J]. Journal of ComputationalPhysics,2013,232(1):397–415.
[34] Zhu J, Zhong X, Shu C W, et al. Runge-Kutta Discontinuous Galerkin MethodUsing a New Type of WENO Limiters on Unstructured Meshes[J]. Journal ofComputational Physics,2013,248:200–220.
[35] Zhang X, Shu C W. On Maximum-Principle-Satisfying High Order Schemesfor Scalar Conservation Laws[J]. Journal of Computational Physics,2010,229(9):3091–3120.
[36] Zhang X, Shu C W. On Positivity-Preserving High Order Discontinuous GalerkinSchemes for Compressible Euler Equations on Rectangular Meshes[J]. Journal ofComputational Physics,2010,229(23):8918–8934.
[37] Zhang X, Shu C W. Positivity-Preserving High Order Discontinuous GalerkinSchemes for Compressible Euler Equations with Source Terms[J]. Journal of Com-putational Physics,2011,230(4):1238–1248.
[38] Zhang X, Xia Y, Shu C W. Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Lawson Triangular Meshes[J]. Journal of Scientific Computing,2012,50(1):29–62.
[39] Zhang Y, Zhang X, Shu C W. Maximum-Principle-Satisfying Second Order Dis-continuous Galerkin Schemes for Convection-Difusion Equations on TriangularMeshes[J]. Journal of Computational Physics,2013,234:295–316.
[40] Xing Y, Zhang X, Shu C W. Positivity-Preserving High Order Well-Balanced Dis-continuous Galerkin Methods for the Shallow Water Equations[J]. Advances inWater Resources,2010,33(12):1476–1493.
[41] Qiu J M, Shu C W. Positivity Preserving Semi-Lagrangian Discontinuous GalerkinFormulation: Theoretical Analysis and Application to the Vlasov–Poisson Sys-tem[J]. Journal of Computational Physics,2011,230(23):8386–8409.
[42] Cheng Y, Li F, Qiu J, et al. Positivity-Preserving DG and Central DG Methods forIdeal MHD Equations[J]. Journal of Computational Physics,2013,238:255–280.
[43] Bassi F, Rebay S. A High-Order Accurate Discontinuous Finite Element Methodfor the Numerical Solution of the Compressible Navier–Stokes Equations[J]. Jour-nal of Computational Physics,1997,131(2):267–279.
[44] Cockburn B, Shu C W. The Local Discontinuous Galerkin Method for Time-Dependent Convection-Difusion Systems[J]. SIAM Journal on Numerical Analy-sis,1998,35(6):2440–2463.
[45] Yan J, Shu C W. A Local Discontinuous Galerkin Method for KdV Type Equation-s[J]. SIAM Journal on Numerical Analysis,2002,40(2):769–791.
[46] Yan J, Shu C W. Local Discontinuous Galerkin Methods for Partial Diferential E-quations with Higher Order Derivatives[J]. Journal of Scientific Computing,2002,17(1):27–47.
[47] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for Two Classes of Two-Dimensional Nonlinear Wave Equations[J]. Physica D,2005,208(1):21–58.
[48] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for the Kuramoto–Sivashinsky Equations and the Ito-Type Coupled KdV Equations[J]. ComputerMethods in Applied Mechanics and Engineering,2006,195(25):3430–3447.
[49] Cockburn B, Shu C W. Runge–Kutta Discontinuous Galerkin Methods forConvection-Dominated Problems[J]. Journal of Scientific Computing,2001,16(3):173–261.
[50] Xu Y, Shu C W. Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Diferential Equations[J]. Communications in ComputationalPhysics,2010,7(1):1–46.
[51] Cheng Y, Shu C W. A Discontinuous Galerkin Finite Element Method for TimeDependent Partial Diferential Equations with Higher Order Derivatives[J]. Math-ematics of Computation,2008,77(262):699–730.
[52] Castillo P, Cockburn B, Perugia I, et al. An a Priori Error Analysis of the Local Dis-continuous Galerkin Method for Elliptic Problems[J]. SIAM Journal on NumericalAnalysis,2000,38(5):1676–1706.
[53] Cockburn B, Gopalakrishnan J, Lazarov R. Unified Hybridization of Discontinu-ous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order EllipticProblems[J]. SIAM Journal on Numerical Analysis,2009,47(2):1319–1365.
[54] Wihler T P, Rivie`re B. Discontinuous Galerkin Methods for Second-Order EllipticPDE with Low-Regularity Solutions[J]. Journal of Scientific Computing,2011,46(2):151–165.
[55] Babusˇka I. The Finite Element Method with Penalty[J]. Mathematics of Compu-tation,1973,27(122):221–228.
[56] Babusˇka I, Zla′mal M. Nonconforming Elements in the Finite Element Methodwith Penalty[J]. SIAM Journal on Numerical Analysis,1973,10(5):863–875.
[57] Baker G A. Finite Element Methods for Elliptic Equations Using NonconformingElements[J]. Mathematics of Computation,1977,31(137):45–59.
[58] Douglas J, Dupont T. Interior Penalty Procedures for Elliptic and ParabolicGalerkin Methods[J]. Computing Methods in Applied Sciences,1976,58:207–216.
[59] Arnold D. An Interior Penalty Finite Element Method with Discontinuous Ele-ments[J]. SIAM Journal on Numerical Analysis,1982,19(4):742–760.
[60] Arnold D, Brezzi F, Cockburn B, et al. Unified Analysis of Discontinuous GalerkinMethods for Elliptic Problems[J]. SIAM Journal on Numerical Analysis,2002,39(5):1749–1779.
[61] Oden J T, Babusˇka I, Baumann C E. A Discontinuous hp Finite Element Methodfor Difusion Problems[J]. Journal of Computational Physics,1998,146(2):491–519.
[62] Gudi T. A New Error Analysis for Discontinuous Finite Element Methods forLinear Elliptic Problems[J]. Mathematics of Computation,2010,79(272):2169–2189.
[63] Rivie`re B. Discontinuous Galerkin Methods for Solving Elliptic and ParabolicEquations: Theory and Implementation[M].1st. Philadelphia: SIAM,2008:1–157.
[64] LeSaint P, Raviart P A. On a Finite Element Method for Solving the NeutronTransport Equation[M].1st. New York: Academic Press,1974:89–145.
[65] Johnson C, Pitka¨ranta J. An Analysis of the Discontinuous Galerkin Method fora Scalar Hyperbolic Equation.[J]. Mathematics of Computation,1986,46(173):1–26.
[66] Peterson T. A Note on the Convergence of the Discontinuous Galerkin Methodfor a Scalar Hyperbolic Equation[J]. SIAM Journal on Numerical Analysis,1991,28(1):133–140.
[67] Richter G R. An Optimal-Order Error Estimate for the Discontinuous GalerkinMethod.[J]. Mathematics of Computation,1988,50(181):75–88.
[68] Cockburn B, Dong B, Guzma′n J. Optimal Convergence of the Original DG Methodfor the Transport-Reaction Equation on Special Meshes[J]. SIAM Journal on Nu-merical Analysis,2008,46(3):1250–1265.
[69] Cockburn B, Dong B, Guzma′n J, et al. Optimal Convergence of the Original DGMethod on Special Meshes for Variable Transport Velocity[J]. SIAM Journal onNumerical Analysis,2010,48(1):133–146.
[70] Zhang Q, Shu C W. Stability Analysis and a Priori Error Estimates of the Third Or-der Explicit Runge-Kutta Discontinuous Galerkin Method for Scalar ConservationLaws[J]. SIAM Journal on Numerical Analysis,2010,48(3):1038–1063.
[71] Zhang Q, Shu C W. Error Estimates to Smooth Solutions of Runge–Kutta Dis-continuous Galerkin Methods for Scalar Conservation Laws[J]. SIAM Journal onNumerical Analysis,2004,42(2):641–666.
[72] Zhang Q, Shu C W. Error Estimates to Smooth Solutions of Runge-Kutta Dis-continuous Galerkin Method for Symmetrizable Systems of Conservation Laws[J].SIAM Journal on Numerical Analysis,2006,44(4):1703–1720.
[73] Cockburn B, Guzma′n J. Error Estimates for the Runge-Kutta DiscontinuousGalerkin Method for the Transport Equation with Discontinuous Initial Data[J].SIAM Journal on Numerical Analysis,2008,46(3):1364–1398.
[74] Zhang Q, Shu C W. Error Estimates for the Third Order Explicit Runge-Kutta Dis-continuous Galerkin Method for Linear Hyperbolic Equation in One-Dimensionwith Discontinuous Initial Data[J]. Numerische Mathematik, In press,2013.
[75] Yang Y, Shu C W. Discontinuous Galerkin Method for Hyperbolic Equations In-volving Delta-Singularities: Negative-Order Norm Error Estimates and Applica-tions[J]. Numerische Mathematik,2013,124(4):753–781.
[76] Jiang G, Shu C W. On a Cell Entropy Inequality for Discontinuous Galerkin Meth-ods[J]. Mathematics of Computation,1994,62(206):531–538.
[77] Cockburn B, Gremaud P A. Error Estimates for Finite Element Methods for ScalarConservation Laws[J]. SIAM Journal on Numerical Analysis,1996,33(2):522–554.
[78] Baccouch M, Adjerid S. Discontinuous Galerkin Error Estimation for HyperbolicProblems on Unstructured Triangular Meshes[J]. Computer Methods in AppliedMechanics and Engineering,2011,200(1):162–177.
[79] Xu Y, Shu C W. Error Estimates of the Semi-Discrete Local Discontinu-ous Galerkin Method for Nonlinear Convection-Difusion and KdV Equations[J].Computer Methods in Applied Mechanics and Engineering,2007,196(37):3805–3822.
[80] Xu Y, Shu C W. Optimal Error Estimates of the Semidiscrete Local DiscontinuousGalerkin Methods for High Order Wave Equations[J]. SIAM Journal on NumericalAnalysis,2012,50(1):79–104.
[81] Cockburn B, Dong B. An Analysis of the Minimal Dissipation Local Discontinu-ous Galerkin Method for Convection-Difusion Problems[J]. Journal of ScientificComputing,2007,32(2):233–262.
[82] Feistauer M, Kucˇera V, Najzar K, et al. Analysis of Space-Time DiscontinuousGalerkin Method for Nonlinear Convection-Difusion Problems[J]. NumerischeMathematik,2011,117(2):251–288.
[83] Dong B, Shu C W. Analysis of a Local Discontinuous Galerkin Method for LinearTime-Dependent Fourth-Order Problems[J]. SIAM Journal on Numerical Analy-sis,2009,47(5):3240–3268.
[84] Bona J L, Chen H, Karakashian O, et al. Conservative Discontinuous GalerkinMethods for the Generalized Korteweg–de Vries Equation[J]. Mathematics ofComputation,2013,82(283):1401–1432.
[85] Adjerid S, Devine K, Flaherty J, et al. A Posteriori Error Estimation for Discontin-uous Galerkin Solutions of Hyperbolic Problems[J]. Computer Methods in AppliedMechanics and Engineering,2002,191(11):1097–1112.
[86] Adjerid S, Massey T. Superconvergence of Discontinuous Galerkin Solutions for aNonlinear Scalar Hyperbolic Problem[J]. Computer Methods in Applied Mechan-ics and Engineering,2006,195(25):3331–3346.
[87] Adjerid S, Klauser A. Superconvergence of Discontinuous Finite Element Solu-tions for Transient Convection-Difusion Problems[J]. Journal of Scientific Com-puting,2005,22(1):5–24.
[88] Huang Q, Xie H, Brunner H. Superconvergence of Discontinuous Galerkin Solu-tions for Delay Diferential Equations of Pantograph Type[J]. SIAM Journal onScientific Computing,2011,33(5):2664–2684.
[89] Celiker F, Cockburn B. Superconvergence of the Numerical Traces of Discontinu-ous Galerkin and Hybridized Methods for Convection-Difusion Problems in OneSpace Dimension[J]. Mathematics of Computation,2007,76(257):67–96.
[90] Cheng Y, Shu C W. Superconvergence and Time Evolution of DiscontinuousGalerkin Finite Element Solutions[J]. Journal of Computational Physics,2008,227(22):9612–9627.
[91] Cheng Y, Shu C W. Superconvergence of Local Discontinuous Galerkin Methodsfor One-Dimensional Convection-Difusion Equations[J]. Computers&Structures,2009,87(11):630–641.
[92] Zhong X, Shu C W. Numerical Resolution of Discontinuous Galerkin Methods forTime Dependent Wave Equations[J]. Computer Methods in Applied Mechanicsand Engineering,2011,200(41):2814–2827.
[93] Cheng Y, Shu C W. Superconvergence of Discontinuous Galerkin and Local Dis-continuous Galerkin Schemes for Linear Hyperbolic and Convection-Difusion E-quations in One Space Dimension[J]. SIAM Journal on Numerical Analysis,2010,47(6):4044–4072.
[94] Huford C, Xing Y. Superconvergence of the Local Discontinuous Galerkin Methodfor the Linearized Korteweg–de Vries Equation[J]. Journal of Computational andApplied Mathematics,2014,255:441–455.
[95] Guo W, Zhong X, Qiu J M. Superconvergence of Discontinuous Galerkin and Lo-cal Discontinuous Galerkin Methods: Eigen-Structure Analysis Based on FourierApproach[J]. Journal of Computational Physics,2013,235:458–485.
[96] Ketcheson D I, Macdonald C B, Gottlieb S. Optimal Implicit Strong Stabili-ty Preserving Runge-Kutta Methods[J]. Applied Numerical Mathematics,2009,59(2):373–392.
[97] Gottlieb S, Ketcheson D I, Shu C W. Strong Stability Preserving Runge-Kutta andMultistep Time Discretizations[M].1st. Singapore: World Scientific,2011:1–164.
[98] Ketcheson D I, Gottlieb S, Macdonald C B. Strong Stability Preserving Two-Step Runge-Kutta Methods[J]. SIAM Journal on Numerical Analysis,2011,49(6):2618–2639.
[99] Xia Y, Xu Y, Shu C W. Efcient Time Discretization for Local DiscontinuousGalerkin Methods[J]. Discrete and Continuous Dynamical Systems–Series B,2007,8(3):677–693.
[100] Christlieb A, Ong B, Qiu J M. Integral Deferred Correction Methods Constructedwith High Order Runge-Kutta Integrators[J]. Mathematics of Computation,2010,79(270):761–783.
[101] Christlieb A, Morton M, Ong B, et al. Semi-Implicit Integral Deferred CorrectionConstructed with Additive Runge-Kutta Methods[J]. Communications in Mathe-matical Sciences,2011,9(3):879–902.
[102] Christlieb A J, Haynes R D, Ong B W. A Parallel Space-Time Algorithm[J]. SIAMJournal on Scientific Computing,2012,34(5):C233–C248.
[103] Guo R, Xu Y. Efcient Solvers of Discontinuous Galerkin Discretization for theCahn–Hilliard Equations[J]. Journal of Scientific Computing, In press,2013.
[104] Su¨li E, Mozolevski I. hp-Version Interior Penalty DGFEMs for the BiharmonicEquation[J]. Computer Methods in Applied Mechanics and Engineering,2007,196(13):1851–1863.
[105] Elliott C M, Zheng S. On the Cahn–Hilliard Equation[J]. Archive for RationalMechanics and Analysis,1986,96(4):339–357.
[106] Feng X, Prohl A. Error Analysis of a Mixed Finite Element Method for the Cahn–Hilliard Equation[J]. Numerische Mathematik,2004,99(1):47–84.
[107] Zhang S, Wang M. A Nonconforming Finite Element Method for the Cahn–Hilliard Equation[J]. Journal of Computational Physics,2010,229(19):7361–7372.
[108] Brenner S C, Gu S, Gudi T, et al. A Quadratic C0Interior Penalty Method forLinear Fourth Order Boundary Value Problems with Boundary Conditions of theCahn–Hilliard Type[J]. SIAM Journal on Numerical Analysis,2012,50(4):2088–2110.
[109] Georgoulis E H, Houston P, Virtanen J. An a Posteriori Error Indicator for Dis-continuous Galerkin Approximations of Fourth-Order Elliptic Problems[J]. IMAJournal of Numerical Analysis,2011,31(1):281–298.
[110] Brenner S, Gudi T, Sung L. An a Posteriori Error Estimator for a Quadratic C0-Interior Penalty Method for the Biharmonic Problem[J]. IMA Journal of NumericalAnalysis,2010,30(3):777–798.
[111] Cockburn B, Dong B, Guzma′n J. A Hybridizable and Superconvergent Discontin-uous Galerkin Method for Biharmonic Problems[J]. Journal of Scientific Comput-ing,2009,40(1):141–187.
[112] Castillo P, Cockburn B, Scho¨tzau D, et al. Optimal a Priori Error Estimates for thehp-Version of the Local Discontinuous Galerkin Method for Convection-DifusionProblems[J]. Mathematics of Computation,2002,71(238):455–478.
[113] Ciarlet P G. The Finite Element Method for Elliptic Problems[M].1st. Amsterdam:North Holland,1978:110–332.
[114] Cockburn B, Luskin M, Shu C W, et al. Enhanced Accuracy by Post-Processingfor Finite Element Methods for Hyperbolic Equations[J]. Mathematics of Compu-tation,2003,72(242):577–606.
[115] Walfisch D, Ryan J K, Kirby R M, et al. One-Sided Smoothness-IncreasingAccuracy-Conserving Filtering for Enhanced Streamline Integration Through Dis-continuous Fields[J]. Journal of Scientific Computing,2009,38(2):164–184.
[116] van Slingerland P, Ryan J K, Vuik C. Position-Dependent Smoothness-IncreasingAccuracy-Conserving (SIAC) Filtering for Improving Discontinuous Galerkin So-lutions[J]. SIAM Journal on Scientific Computing,2011,33(2):802–825.
[117] Mirzaee H, Ryan J K, Kirby R M. Efcient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin So-lutions[J]. Journal of Scientific Computing,2012,52(1):85–112.
[118] King J, Mirzaee H, Ryan J K, et al. Smoothness-Increasing Accuracy-Conserving(SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors versusHigher-Order Accuracy[J]. Journal of Scientific Computing,2012,53(1):129–149.
[119] Mirzaee H, Ji L, Ryan J K, et al. Smoothness-Increasing Accuracy-Conserving(SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Tri-angular Meshes[J]. SIAM Journal on Numerical Analysis,2011,49(5):1899–1920.
[120] Ji L, Xu Y, Ryan J K. Negative-Order Norm Estimates for Nonlinear HyperbolicConservation Laws[J]. Journal of Scientific Computing,2013,54(2-3):531–548.
[121] Ji L, Xu Y, Ryan J K. Accuracy-Enhancement of Discontinuous Galerkin Solutionsfor Convection-Difusion Equations in Multiple-Dimensions[J]. Mathematics ofComputation,2012,81(280):1929–1950.
[122] LeVeque R. Numerical Methods for Conservation Laws[M].2nd. Basel:Birkha¨user Verlag,2005:97–207.
[123] Brenner S, Scott L R. The Mathematical Theory of Finite Element Methods[M].3rd. New York: Springer,2007:105–110.
[124] Gottlieb S, Ketcheson D I, Shu C W. High Order Strong Stability Preserving TimeDiscretizations[J]. Journal of Scientific Computing,2009,38(3):251–289.
[125] Cockburn B, Kanschat G, Perugia I, et al. Superconvergence of the Local Dis-continuous Galerkin Method for Elliptic Problems on Cartesian Grids[J]. SIAMJournal on Numerical Analysis,2001,39(1):264–285.