Top

Journal of Scientific Computing

Published in:

01-05-2023

On Generalized Gauss–Radau Projections and Optimal Error Estimates of Upwind-Biased DG Methods for the Linear Advection Equation on Special Simplex Meshes

Authors: Zheng Sun, Yulong Xing

Published in: Journal of Scientific Computing | Issue 2/2023

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Generalized Gauss–Radau (GGR) projections are global projection operators that are widely used for the error analysis of discontinuous Galerkin (DG) methods with generalized numerical fluxes. In previous work, GGR projections were constructed for Cartesian meshes and analyzed through an algebraic approach. In this paper, we first present an alternative energy approach for analyzing the one-dimensional GGR projection, which does not require assembling and explicitly solving a global system over the entire computational domain as that in the algebraic approach. We then generalize this energy argument to construct a global projection operator on special simplex meshes in multidimensions satisfying the so-called flow condition. With this projection, optimal error estimates are proved for upwind-biased DG methods for the linear advection equation on these meshes, which generalizes the error analysis for the purely upwind case by Cockburn et al. (SIAM J Numer Anal 46(3):1250–1265, 2008) in a time-dependent setting.

previous article Proximal Quasi-Newton Method for Composite Optimization over the Stiefel Manifold

next article Solving Nonlinear Elliptic Inverse Source, Coefficient and Conductivity Problems by the Methods with Bases Satisfying the Boundary Conditions Automatically

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Since \({\left| \left| \left| \cdot \right| \right| \right| }\) is already a seminorm, it suffices to show \(|v(1)| = 0\) implies \(v\equiv 0\), \(\forall v \in P_{k-1}^\perp (\hat{I})\). Indeed, note that \(P_{k-1}^\perp (\hat{I}) = \left\{ al_k(x)|a\in \mathbb {R}\right\} \), where \(l_k(x)\) is the kth-order Legendre polynomial on \(\hat{I}\). For \(v = a l_k(x)\), since \(l_k(1) \ne 0\), one can see that \(v(1) = 0\) implies \(a = 0\) and hence \(v \equiv 0\).

In the papers by Cockburn et al., the estimate of \(\Vert u - \Pi _1 u \Vert _{L^2(K)}\) is proved. The estimate of the trace \(\Vert u - \Pi _1 u \Vert _{L^2\left( e_K^+\right) }\) can be obtained after applying the inverse trace inequality.

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefMATH

Bona, J., Chen, H., Karakashian, O., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)MathSciNetCrossRefMATH

Brandts, J., Korotov, S., Křížek, M.: On the equivalence of regularity criteria for triangular and tetrahedral finite element partitions. Comput. Math. Appl. 55(10), 2227–2233 (2008)MathSciNetMATH

Castillo, P., Cockburn, B., Schötzau, D., Schwab, C.: Optimal a priori error estimates for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455–478 (2002)MathSciNetCrossRefMATH

Cheng, Y., Chou, C.-S., Li, F., Xing, Y.: \({L}^2\) stable discontinuous Galerkin methods for one-dimensional two-way wave equations. Math. Comput. 86(303), 121–155 (2017)CrossRefMATH

Cheng, Y., Meng, X., Zhang, Q.: Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations. Math. Comput. 86(305), 1233–1267 (2017)MathSciNetCrossRefMATH

Cheng, Y., Zhang, Q.: Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem. J. Sci. Comput. 72(2), 792–819 (2017)MathSciNetCrossRefMATH

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland (1978)

Cockburn, B., Dong, B., Guzmán, J.: Optimal convergence of the original DG method for the transport-reaction equation on special meshes. SIAM J. Numer. Anal. 46(3), 1250–1265 (2008)MathSciNetCrossRefMATH

10.

Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887–1916 (2008)MathSciNetCrossRefMATH

11.

Cockburn, B., Dong, B., Guzmán, J., Qian, J.: Optimal convergence of the original DG method on special meshes for variable transport velocity. SIAM J. Numer. Anal. 48(1), 133–146 (2010)MathSciNetCrossRefMATH

12.

Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79(271), 1351–1367 (2010)MathSciNetCrossRefMATH

13.

Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39(1), 264–285 (2001)MathSciNetCrossRefMATH

14.

Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin methods: Theory, Computation and Applications, vol. 11. Springer Science & Business Media (2012)

15.

Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)MathSciNetCrossRefMATH

16.

Dong, B.: Optimally convergent hdg method for third-order Korteweg-de Vries type equations. J. Sci. Comput. 73(2), 712–735 (2017)MathSciNetCrossRefMATH

17.

Fu, G., Shu, C.-W.: Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems. J. Comput. Phys. 394(1), 329–363 (2019)MathSciNetCrossRefMATH

18.

Fu, P., Cheng, Y., Li, F., Xu, Y.: Discontinuous Galerkin methods with optimal \({L}^{2}\) accuracy for one dimensional linear PDEs with high order spatial derivatives. J. Sci. Comput. 78(2), 816–863 (2019)MathSciNetCrossRefMATH

19.

Li, J., Zhang, D., Meng, X., Wu, B.: Analysis of discontinuous Galerkin methods with upwind-biased fluxes for one dimensional linear hyperbolic equations with degenerate variable coefficients. J. Sci. Comput. 78(3), 1305–1328 (2019)MathSciNetCrossRefMATH

20.

Li, J., Zhang, D., Meng, X., Wu, B.: Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations. Math. Comput. 89(325), 2085–2111 (2020)MathSciNetCrossRefMATH

21.

Li, J., Zhang, D., Meng, X., Wu, B., Zhang, Q.: Discontinuous Galerkin methods for nonlinear scalar conservation laws: Generalized local Lax-Friedrichs numerical fluxes. SIAM J. Numer. Anal. 58(1), 1–20 (2020)MathSciNetCrossRefMATH

22.

Liu, M., Wu, B., Meng, X.: Optimal error estimates of the discontinuous Galerkin method with upwind-biased fluxes for 2d linear variable coefficients hyperbolic equations. J. Sci. Comput. 83(1), 1–19 (2020)MathSciNetCrossRefMATH

23.

Liu, Y., Shu, C.-W., Zhang, M.: Optimal error estimates of the semidiscrete central discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 56(1), 520–541 (2018)MathSciNetCrossRefMATH

24.

Liu, Y., Shu, C.-W., Zhang, M.: Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using \({P}^k\) elements. ESAIM Math. Modell. Numer. Anal., 54, 705–726 (2020)

25.

Liu, Y., Shu, C.-W., Zhang, M.: Sub-optimal convergence of discontinuous Galerkin methods with central fluxes for linear hyperbolic equations with even degree polynomial approximations. J. Comput. Math. 39(4), 518–537 (2021)MathSciNetMATH

26.

Meng, X., Shu, C.-W., Wu, B.: Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85(299), 1225–1261 (2016)MathSciNetCrossRefMATH

27.

Peterson, T.E.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28(1), 133–140 (1991)MathSciNetCrossRefMATH

28.

Reed, W.H., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical report, Los Alamos Scientific Lab., N. Mex. (USA) (1973)

29.

Schöberl, J.: NETGEN an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1(1), 41–52 (1997)CrossRefMATH

30.

Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. Numer. Solutions Partial Different. Eqn., 201 (2009)

31.

Sun, J., Shu, C.-W., Xing, Y.: Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise. ESAIM Math. Modell. Numer. Anal. (2022). https://doi.org/10.1051/m2an/2022084

32.

Sun, J., Shu, C.-W., Xing, Y.: Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise. J. Comput. Phys. 461, 111199 (2022)MathSciNetCrossRefMATH

33.

Sun, Z., Shu, C.-W.: Stability of the fourth order Runge-Kutta method for time-dependent partial differential equations. Ann. Math. Sci. Appl. 2(2), 255–284 (2017)MathSciNetCrossRefMATH

34.

Sun, Z., Shu, C.-W.: Strong stability of explicit Runge-Kutta time discretizations. SIAM J. Numer. Anal. 57(3), 1158–1182 (2019)MathSciNetCrossRefMATH

35.

Sun, Z., Shu, C.-W.: Error analysis of Runge–Kutta discontinuous Galerkin methods for linear time-dependent partial differential equations. arXiv preprint arXiv:2001.00971 (2020)

36.

Sun, Z., Xing, Y.: Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes. Math. Comput. 90(330), 1741–1772 (2021)MathSciNetCrossRefMATH

37.

Wang, H., Zhang, Q., Shu, C.-W.: Implicit-explicit local discontinuous Galerkin methods with generalized alternating numerical fluxes for convection-diffusion problems. J. Sci. Comput. 81(3), 2080–2114 (2019)MathSciNetCrossRefMATH

38.

Xu, Y., Shu, C.-W., Zhang, Q.: Error estimate of the fourth-order Runge-Kutta discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 58(5), 2885–2914 (2020)MathSciNetCrossRefMATH

39.

Xu, Y., Zhang, Q., Shu, C.-W., Wang, H.: The \({L} ^2\)-norm stability analysis of Runge-Kutta discontinuous Galerkin methods for linear hyperbolic equations. SIAM J. Numer. Anal. 57(4), 1574–1601 (2019)MathSciNetCrossRefMATH

40.

Xu, Y., Zhao, D., Zhang, Q.: Local error estimates for Runge-Kutta discontinuous Galerkin methods with upwind-biased numerical fluxes for a linear hyperbolic equation in one-dimension with discontinuous initial data. J. Sci. Comput. 91(1), 1–30 (2022)MathSciNetCrossRefMATH

41.

Zhang, H., Wu, B., Meng, X.: A local discontinuous Galerkin method with generalized alternating fluxes for 2D nonlinear Schrödinger equations. Commun. Appl. Math. Comput. 4(1), 84–107 (2022)MathSciNetCrossRefMATH

42.

Zhang, H., Wu, B., Meng, X.: Analysis of the local discontinuous galerkin method with generalized fluxes for one-dimensional nonlinear convection-diffusion systems. Science China Math. 65 (2022). https://doi.org/10.1007/s11425-022-2035-y

43.

Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimates of the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48(3), 1038–1063 (2010)MathSciNetCrossRefMATH

44.

Zhao, D., Zhang, Q.: Local discontinuous Galerkin methods with generalized alternating numerical fluxes for two-dimensional linear Sobolev equation. J. Sci. Comput. 78(3), 1660–1690 (2019)MathSciNetCrossRefMATH

Title: On Generalized Gauss–Radau Projections and Optimal Error Estimates of Upwind-Biased DG Methods for the Linear Advection Equation on Special Simplex Meshes
Authors: Zheng Sun
Yulong Xing
Publication date: 01-05-2023
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2023
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-023-02166-w

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 2/2023

Practical Sketching Algorithms for Low-Rank Tucker Approximation of Large Tensors

Superconvergence Analysis of Curlcurl-Conforming Elements on Rectangular Meshes

Proximal Quasi-Newton Method for Composite Optimization over the Stiefel Manifold

Accuracy and Architecture Studies of Residual Neural Network Method for Ordinary Differential Equations

Quasi Non-Negative Quaternion Matrix Factorization with Application to Color Face Recognition

Analysis and Hermite Spectral Approximation of Diffusive-Viscous Wave Equations in Unbounded Domains Arising in Geophysics

Premium Partner