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Volumn 7, Issue 1, 2010, Pages 474-508

Local discontinuous galerkin methods for high-order time-dependent partial differential equations

Author keywords

Discontinuous Galerkin method; Error estimates; High order accuracy; Local discontinuous Galerkin method; Numerical flux; Stability; Time discretization

Indexed keywords


EID: 76449116016     PISSN: 18152406     EISSN: 19917120     Source Type: Journal    
DOI: 10.4208/cicp.2009.09.023     Document Type: Review
Times cited : (225)

References (109)
  • 1
    • 35348832931 scopus 로고    scopus 로고
    • A discontinuous Galerkin method for higher-order ordinary differential equations
    • S. Adjerid and H. Temimi. A discontinuous Galerkin method for higher-order ordinary differential equations. Comput. Meth. Appl. Mech. Eng., 197:202-218, 2007.
    • (2007) Comput. Meth. Appl. Mech. Eng , vol.197 , pp. 202-218
    • Adjerid, S.1    Temimi, H.2
  • 3
    • 33750455856 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for three-dimensional shallowwater flow
    • V. Aizinger and C. Dawson. The local discontinuous Galerkin method for three-dimensional shallowwater flow. Comput.Methods Appl.Mech. Engrg., 196:734-746, 2007.
    • (2007) Comput.Methods Appl.Mech. Engrg , vol.196 , pp. 734-746
    • Aizinger, V.1    Dawson, C.2
  • 5
  • 6
    • 0001163226 scopus 로고
    • An interior penalty finite element method with discontinuous elements
    • D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19:742-760, 1982.
    • (1982) SIAM J. Numer. Anal , vol.19 , pp. 742-760
    • Arnold, D.N.1
  • 7
    • 0036410391 scopus 로고    scopus 로고
    • Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAMJ
    • D. N. Arnold, F. Brezzi, B. Cockburn, and L. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAMJ. Numer. Anal., 39:1749-1779, 2002.
    • (2002) Numer. Anal , vol.39 , pp. 1749-1779
    • Arnold, D.N.1    Brezzi, F.2    Cockburn, B.3    Marini, L.4
  • 9
    • 84904152686 scopus 로고
    • Finite element methods for elliptic equations using nonconforming elements
    • G. Baker. Finite element methods for elliptic equations using nonconforming elements. Math. Comp., 31:45-59, 1977.
    • (1977) Math. Comp , vol.31 , pp. 45-59
    • Baker, G.1
  • 10
    • 0001416032 scopus 로고    scopus 로고
    • A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations
    • F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131:267-279, 1997.
    • (1997) J. Comput. Phys , vol.131 , pp. 267-279
    • Bassi, F.1    Rebay, S.2
  • 11
    • 0345062335 scopus 로고    scopus 로고
    • A discontinuous hp finite element method for convection-diffusion problems
    • C. Baumann and J. Oden. A discontinuous hp finite element method for convection-diffusion problems. Comput. Meth. Appl. Mech. Eng., 175:311-341, 1999.
    • (1999) Comput. Meth. Appl. Mech. Eng , vol.175 , pp. 311-341
    • Baumann, C.1    Oden, J.2
  • 12
    • 0028407037 scopus 로고
    • Parallel, adaptive finite element methods for conservation laws
    • R. Biswas, K. Devine, and J. Flaherty. Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14:255-283, 1994.
    • (1994) Appl. Numer. Math , vol.14 , pp. 255-283
    • Biswas, R.1    Devine, K.2    Flaherty, J.3
  • 13
    • 61349173997 scopus 로고    scopus 로고
    • Local discontinuous Galerkinmethod with reduced stabilization for diffusion equations
    • E. Burman and B. Stamm. Local discontinuous Galerkinmethod with reduced stabilization for diffusion equations. Commun. Comput. Phys., 5:498-514, 2009.
    • (2009) Commun. Comput. Phys , vol.5 , pp. 498-514
    • Burman, E.1    Stamm, B.2
  • 14
    • 33745961680 scopus 로고    scopus 로고
    • A unified analysis of the local discontinuous Galerkin method for a class of nonlinear problems
    • R. Bustinza. A unified analysis of the local discontinuous Galerkin method for a class of nonlinear problems. Appl. Numer. Math., 56:1293-1306, 2006.
    • (2006) Appl. Numer. Math , vol.56 , pp. 1293-1306
    • Bustinza, R.1
  • 15
    • 14544305193 scopus 로고    scopus 로고
    • A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAMJ
    • R. Bustinza and G. Gatica. A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. SIAMJ. Sci. Comput., 26:152-177, 2004.
    • (2004) Sci. Comput , vol.26 , pp. 152-177
    • Bustinza, R.1    Gatica, G.2
  • 16
    • 18844446713 scopus 로고    scopus 로고
    • A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions
    • R. Bustinza and G. Gatica. A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions. J. Comput. Phys., 207:427-456, 2005.
    • (2005) J. Comput. Phys , vol.207 , pp. 427-456
    • Bustinza, R.1    Gatica, G.2
  • 17
    • 17444369021 scopus 로고    scopus 로고
    • An A posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems
    • R. Bustinza, G. Gatica, and B. Cockburn. An A posteriori error estimate for the local discontinuous Galerkin method applied to linear and nonlinear diffusion problems. J. Sci. Comput., 22-23:147-185, 2005.
    • (2005) J. Sci. Comput , vol.22-23 , pp. 147-185
    • Bustinza, R.1    Gatica, G.2    Cockburn, B.3
  • 18
    • 41549122815 scopus 로고    scopus 로고
    • On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems
    • R. Bustinza, G. Gatica, and F. Sayas. On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems. IMA J. Numer. Anal., 28:225-244, 2008.
    • (2008) IMA J. Numer. Anal , vol.28 , pp. 225-244
    • Bustinza, R.1    Gatica, G.2    Sayas, F.3
  • 19
    • 17444400815 scopus 로고    scopus 로고
    • An a posteriori error estimate for the local discontinuous Galerkin method
    • P. Castillo. An a posteriori error estimate for the local discontinuous Galerkin method. J. Sci. Comput., 22-23:187-204, 2005.
    • (2005) J. Sci. Comput , vol.22-23 , pp. 187-204
    • Castillo, P.1
  • 20
    • 33746172644 scopus 로고    scopus 로고
    • A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems
    • P. Castillo. A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems. Appl. Numer. Math., 56:1307-1313, 2006.
    • (2006) Appl. Numer. Math , vol.56 , pp. 1307-1313
    • Castillo, P.1
  • 21
    • 0034548629 scopus 로고    scopus 로고
    • An a priori error analysis of the local discontinuous Galerkinmethod for elliptic problems. SIAMJ
    • P. Castillo, B. Cockburn, I. Perugia, and D. Schotzau. An a priori error analysis of the local discontinuous Galerkinmethod for elliptic problems. SIAMJ. Numer. Anal., 38:1676-1706, 2000.
    • (2000) Numer. Anal , vol.38 , pp. 1676-1706
    • Castillo, P.1    Cockburn, B.2    Perugia, I.3    Schotzau, D.4
  • 23
    • 42149105254 scopus 로고    scopus 로고
    • A numerical study of a semi-algebraic multilevel preconditioner for the local discontinuous Galerkin method
    • P. Castillo and E. Velazquez. A numerical study of a semi-algebraic multilevel preconditioner for the local discontinuous Galerkin method. Int. J. Numer. Methods Eng., 74:255-268, 2008.
    • (2008) Int. J. Numer. Methods Eng , vol.74 , pp. 255-268
    • Castillo, P.1    Velazquez, E.2
  • 24
    • 21644478120 scopus 로고    scopus 로고
    • Pointwise error estimates of the local discontinuous Galerkinmethod for a second order elliptic problem
    • H. Chen. Pointwise error estimates of the local discontinuous Galerkinmethod for a second order elliptic problem. Math. Comp., 74:1097-1116, 2005.
    • (2005) Math. Comp , vol.74 , pp. 1097-1116
    • Chen, H.1
  • 25
    • 42349102247 scopus 로고    scopus 로고
    • A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives
    • Y. Cheng and C.-W. Shu. A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comp., 77:699-730, 2008.
    • (2008) Math. Comp , vol.77 , pp. 699-730
    • Cheng, Y.1    Shu, C.-W.2
  • 26
    • 76449109073 scopus 로고    scopus 로고
    • B. Cockburn. Discontinuous Galerkin methods for convection-dominated problems. In T. Barth and H. Deconinck, editors, High-Order Methods for Computational Physics, 9 of LectureNotes in Computational Science and Engineering, pages 69-224. Springer, Berlin, 1999.
    • B. Cockburn. Discontinuous Galerkin methods for convection-dominated problems. In T. Barth and H. Deconinck, editors, High-Order Methods for Computational Physics, volume 9 of LectureNotes in Computational Science and Engineering, pages 69-224. Springer, Berlin, 1999.
  • 27
    • 34547254704 scopus 로고    scopus 로고
    • An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems
    • B. Cockburn and B. Dong. An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput., 32:233-262, 2007.
    • (2007) J. Sci. Comput , vol.32 , pp. 233-262
    • Cockburn, B.1    Dong, B.2
  • 28
    • 0002154319 scopus 로고    scopus 로고
    • Error estimates for finite element methods for nonlinear conservation laws
    • B. Cockburn and P. Gremaud. Error estimates for finite element methods for nonlinear conservation laws. SIAM J. Numer. Anal., 33:522-554, 1996.
    • (1996) SIAM J. Numer. Anal , vol.33 , pp. 522-554
    • Cockburn, B.1    Gremaud, P.2
  • 29
    • 84966261380 scopus 로고
    • The Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws IV: Themultidimensional case
    • B. Cockburn, S. Hou, and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws IV: themultidimensional case. Math. Comp., 54:545-581, 1990.
    • (1990) Math. Comp , vol.54 , pp. 545-581
    • Cockburn, B.1    Hou, S.2    Shu, C.-W.3
  • 30
    • 1642602854 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for the Oseen equations
    • B. Cockburn, G. Kanschat, and D. Schotzau. The local discontinuous Galerkin method for the Oseen equations. Math. Comp., 73:569-593, 2004.
    • (2004) Math. Comp , vol.73 , pp. 569-593
    • Cockburn, B.1    Kanschat, G.2    Schotzau, D.3
  • 31
    • 21644481738 scopus 로고    scopus 로고
    • A locally conservative LDG method for the incompressible Navier-Stokes equations
    • B. Cockburn, G. Kanschat, and D. Schotzau. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp., 74:1067-1095, 2005.
    • (2005) Math. Comp , vol.74 , pp. 1067-1095
    • Cockburn, B.1    Kanschat, G.2    Schotzau, D.3
  • 32
    • 11144289535 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for linearized incompressible fluid flow: A review
    • B. Cockburn, G. Kanschat, and D. Schotzau. The local discontinuous Galerkin method for linearized incompressible fluid flow: a review. Comput. &Fluids, 34:491-506, 2005.
    • (2005) Comput. &Fluids , vol.34 , pp. 491-506
    • Cockburn, B.1    Kanschat, G.2    Schotzau, D.3
  • 34
    • 0002084927 scopus 로고    scopus 로고
    • The development of discontinuous Galerkin methods
    • B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, of, Springer, Berlin
    • B. Cockburn, G. Karniadakis, and C.-W. Shu. The development of discontinuous Galerkin methods. In B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 of Lecture Notes in Computational Science and Engineering, pages 3-50. Springer, Berlin, 2000.
    • (2000) Lecture Notes in Computational Science and Engineering , vol.11 , pp. 3-50
    • Cockburn, B.1    Karniadakis, G.2    Shu, C.-W.3
  • 35
    • 45249130080 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws III: One dimensional systems
    • B. Cockburn, S.-Y. Lin, and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws III: one dimensional systems. J. Comput. Phys., 84:90-113, 1989.
    • (1989) J. Comput. Phys , vol.84 , pp. 90-113
    • Cockburn, B.1    Lin, S.-Y.2    Shu, C.-W.3
  • 36
    • 0037376067 scopus 로고    scopus 로고
    • Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
    • B. Cockburn, M. Luskin, C.-W. Shu, and E. Süli. Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comp., 72:577-606, 2003.
    • (2003) Math. Comp , vol.72 , pp. 577-606
    • Cockburn, B.1    Luskin, M.2    Shu, C.-W.3    Süli, E.4
  • 37
    • 84966246413 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework
    • B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comp., 52:411-435, 1989.
    • (1989) Math. Comp , vol.52 , pp. 411-435
    • Cockburn, B.1    Shu, C.-W.2
  • 38
    • 0344393366 scopus 로고    scopus 로고
    • The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAMJ
    • B. Cockburn and C.-W. Shu. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAMJ. Numer. Anal., 35:2440-2463, 1998.
    • (1998) Numer. Anal , vol.35 , pp. 2440-2463
    • Cockburn, B.1    Shu, C.-W.2
  • 39
    • 0001690553 scopus 로고    scopus 로고
    • The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems
    • B. Cockburn and C.-W. Shu. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys., 141:199-224, 1998.
    • (1998) J. Comput. Phys , vol.141 , pp. 199-224
    • Cockburn, B.1    Shu, C.-W.2
  • 40
    • 0000076512 scopus 로고    scopus 로고
    • Runge-Kutta discontinuous Galerkin methods for convection-dominated problems
    • B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 16:173-261, 2001.
    • (2001) J. Sci. Comput , vol.16 , pp. 173-261
    • Cockburn, B.1    Shu, C.-W.2
  • 41
    • 29144482223 scopus 로고    scopus 로고
    • Foreword for the special issue on discontinuous Galerkin method
    • B. Cockburn and C.-W. Shu. Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput., 22-23:1-3, 2005.
    • (2005) J. Sci. Comput , vol.22-23 , pp. 1-3
    • Cockburn, B.1    Shu, C.-W.2
  • 42
    • 67349193309 scopus 로고    scopus 로고
    • Foreword for the special issue on discontinuous Galerkin method
    • B. Cockburn and C.-W. Shu. Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput., 40:1-3, 2009.
    • (2009) J. Sci. Comput , vol.40 , pp. 1-3
    • Cockburn, B.1    Shu, C.-W.2
  • 43
    • 0036501083 scopus 로고    scopus 로고
    • Exponential time differencing for stiff systems
    • S. Cox and P. Matthews. Exponential time differencing for stiff systems. J. Comput. Phys., 176:430-455, 2002.
    • (2002) J. Comput. Phys , vol.176 , pp. 430-455
    • Cox, S.1    Matthews, P.2
  • 44
    • 1642503911 scopus 로고    scopus 로고
    • k local discontinuous Galerkin method for elliptic equations
    • k local discontinuous Galerkin method for elliptic equations. SIAM J. Numer. Anal., 40:2151-2170, 2003.
    • (2003) SIAM J. Numer. Anal , vol.40 , pp. 2151-2170
    • Dawson, C.1
  • 45
    • 19944366692 scopus 로고    scopus 로고
    • Coupling local discontinuous and continuous Galerkinmethods for flow problems
    • C. Dawson. Coupling local discontinuous and continuous Galerkinmethods for flow problems. Advances inWater Resources, 28:729-744, 2005.
    • (2005) Advances inWater Resources , vol.28 , pp. 729-744
    • Dawson, C.1
  • 46
    • 33646078908 scopus 로고    scopus 로고
    • Foreword for the special issue on discontinuous Galerkin method
    • Engin
    • C. Dawson. Foreword for the special issue on discontinuous Galerkin method. Comput. Meth. Appl. Mech. Engin., 195:3183, 2006.
    • (2006) Comput. Meth. Appl. Mech , vol.195 , pp. 3183
    • Dawson, C.1
  • 47
    • 0035499968 scopus 로고    scopus 로고
    • A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations
    • C. Dawson and J. Proft. A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations. Numerical Methods for Partial Differential Equations, 17:545-564, 2001.
    • (2001) Numerical Methods for Partial Differential Equations , vol.17 , pp. 545-564
    • Dawson, C.1    Proft, J.2
  • 48
    • 76449111912 scopus 로고    scopus 로고
    • Analysis of a local discontinuous Galerkinmethod for fourth-order time-dependent problems. SIAMJ
    • submitted
    • B. Dong and C.-W. Shu. Analysis of a local discontinuous Galerkinmethod for fourth-order time-dependent problems. SIAMJ. Numer. Anal. submitted.
    • Numer. Anal
    • Dong, B.1    Shu, C.-W.2
  • 49
    • 0042185232 scopus 로고    scopus 로고
    • Spectral deferred correction methods for ordinary differential equations
    • A. Dutt, L. Greengard, and V. Rokhlin. Spectral deferred correction methods for ordinary differential equations. BIT, 40:241-266, 2000.
    • (2000) BIT , vol.40 , pp. 241-266
    • Dutt, A.1    Greengard, L.2    Rokhlin, V.3
  • 50
    • 1942507896 scopus 로고    scopus 로고
    • A discontinuous spectral element model for Boussinesq-type equations
    • C. Eskilsson and S. Sherwin. A discontinuous spectral element model for Boussinesq-type equations. J. Sci. Comput., 17:143-152, 2002.
    • (2002) J. Sci. Comput , vol.17 , pp. 143-152
    • Eskilsson, C.1    Sherwin, S.2
  • 51
    • 14844358935 scopus 로고    scopus 로고
    • Discontinuous Galerkin spectral/hp element modelling of dispersive shallow water systems
    • C. Eskilsson and S. Sherwin. Discontinuous Galerkin spectral/hp element modelling of dispersive shallow water systems. J. Sci. Comput., 22/23:269-288, 2005.
    • (2005) J. Sci. Comput , vol.22-23 , pp. 269-288
    • Eskilsson, C.1    Sherwin, S.2
  • 52
    • 29144479606 scopus 로고    scopus 로고
    • Spectral/hp discontinuous Galerkin methods for modeling 2D Boussinesq equations
    • C. Eskilsson and S. Sherwin. Spectral/hp discontinuous Galerkin methods for modeling 2D Boussinesq equations. J. Comput. Phys., 212:566-589, 2006.
    • (2006) J. Comput. Phys , vol.212 , pp. 566-589
    • Eskilsson, C.1    Sherwin, S.2
  • 53
    • 0036680851 scopus 로고    scopus 로고
    • J. E. Flaherty, L. Krivodonova, J.-F. Remacle, andM. S. Shephard. Aspects of discontinuous Galerkin methods for hyperbolic conservation laws. Finite Elem. Anal. Des., 38:889-908, 2002.
    • J. E. Flaherty, L. Krivodonova, J.-F. Remacle, andM. S. Shephard. Aspects of discontinuous Galerkin methods for hyperbolic conservation laws. Finite Elem. Anal. Des., 38:889-908, 2002.
  • 54
    • 34248593303 scopus 로고    scopus 로고
    • A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes
    • G. Gassner, F. Lörcher, and C.-D. Munz. A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys., 224:1049-1063, 2007.
    • (2007) J. Comput. Phys , vol.224 , pp. 1049-1063
    • Gassner, G.1    Lörcher, F.2    Munz, C.-D.3
  • 55
    • 0032345207 scopus 로고    scopus 로고
    • Total variation diminishing Runge-Kutta schemes
    • S. Gottlieb and C.-W. Shu. Total variation diminishing Runge-Kutta schemes. Math. Comp., 67:73-85, 1998.
    • (1998) Math. Comp , vol.67 , pp. 73-85
    • Gottlieb, S.1    Shu, C.-W.2
  • 56
    • 67349146518 scopus 로고    scopus 로고
    • Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations
    • J. Guzmán and B. Rivière. Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations. J. Sci. Comput., 40:273-280, 2009.
    • (2009) J. Sci. Comput , vol.40 , pp. 273-280
    • Guzmán, J.1    Rivière, B.2
  • 57
    • 54449085907 scopus 로고    scopus 로고
    • Local discontinuous-Galerkin schemes for model problems in phase transition theory
    • J. Haink and C. Rohde. Local discontinuous-Galerkin schemes for model problems in phase transition theory. Commun. Comput. Phys., 4:860-893, 2008.
    • (2008) Commun. Comput. Phys , vol.4 , pp. 860-893
    • Haink, J.1    Rohde, C.2
  • 59
    • 0000182351 scopus 로고    scopus 로고
    • Stabilized hp-?nite element methods for hyperbolic problems. SIAMJ
    • P. Houston, E. Süli, and C. Schwab. Stabilized hp-?nite element methods for hyperbolic problems. SIAMJ. Numer. Anal., 37:1618-1643, 2001.
    • (2001) Numer. Anal , vol.37 , pp. 1618-1643
    • Houston, P.1    Süli, E.2    Schwab, C.3
  • 60
    • 84968487025 scopus 로고
    • On a cell entropy inequality for discontinuous Galerkin methods
    • G. Jiang and C.-W. Shu. On a cell entropy inequality for discontinuous Galerkin methods. Math. Comp., 62:531-538, 1994.
    • (1994) Math. Comp , vol.62 , pp. 531-538
    • Jiang, G.1    Shu, C.-W.2
  • 61
    • 2942574812 scopus 로고    scopus 로고
    • Preconditioning methods for local discontinuous Galerkin discretizations
    • G. Kanschat. Preconditioning methods for local discontinuous Galerkin discretizations. SIAM J. Sci. Comput., 25:815-831, 2003.
    • (2003) SIAM J. Sci. Comput , vol.25 , pp. 815-831
    • Kanschat, G.1
  • 62
    • 17444402857 scopus 로고    scopus 로고
    • Block preconditioners for LDG discretizations of linear incompressible flow problems
    • G. Kanschat. Block preconditioners for LDG discretizations of linear incompressible flow problems. J. Sci. Comput., 22-23:371-384, 2005.
    • (2005) J. Sci. Comput , vol.22-23 , pp. 371-384
    • Kanschat, G.1
  • 64
    • 22544474620 scopus 로고    scopus 로고
    • Fourth-order time stepping for stiff problem
    • A.-K. Kassam and L. Trefethen. Fourth-order time stepping for stiff problem. SIAM J. Sci. Comput., 26:1214-1233, 2005.
    • (2005) SIAM J. Sci. Comput , vol.26 , pp. 1214-1233
    • Kassam, A.-K.1    Trefethen, L.2
  • 65
    • 0037216570 scopus 로고    scopus 로고
    • Additive Runge-Kutta schemes for convection-diffusion-reaction equations
    • C. Kennedy and M. Carpenter. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Num. Math., 44:139-181, 2003.
    • (2003) Appl. Num. Math , vol.44 , pp. 139-181
    • Kennedy, C.1    Carpenter, M.2
  • 66
    • 0000451632 scopus 로고    scopus 로고
    • A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems
    • B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, of, Springer
    • M. Larson and T. Barth. A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems. In B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 of Lecture Notes in Computational Science and Engineering, pages 221-230. Springer, 2000.
    • (2000) Lecture Notes in Computational Science and Engineering , vol.11 , pp. 221-230
    • Larson, M.1    Barth, T.2
  • 67
    • 2442475431 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for nonlinear dispersive equations
    • D. Levy, C.-W. Shu, and J. Yan. Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys., 196:751-772, 2004.
    • (2004) J. Comput. Phys , vol.196 , pp. 751-772
    • Levy, D.1    Shu, C.-W.2    Yan, J.3
  • 69
    • 33846373762 scopus 로고    scopus 로고
    • Local discontinuous Galerkin approximations to Richards' equation
    • H. Li,M. Farthing, C. Dawson, and C.Miller. Local discontinuous Galerkin approximations to Richards' equation. Advances inWater Resources, 30:555-575, 2007.
    • (2007) Advances inWater Resources , vol.30 , pp. 555-575
    • Li, H.1    Farthing, M.2    Dawson, C.3    Miller, C.4
  • 70
    • 34250652075 scopus 로고    scopus 로고
    • Adaptive local discontinuous Galerkin approximation to Richards' equation
    • H. Li, M. Farthing, and C.Miller. Adaptive local discontinuous Galerkin approximation to Richards' equation. Advances in Water Resources, 30:1883-1901, 2007.
    • (2007) Advances in Water Resources , vol.30 , pp. 1883-1901
    • Li, H.1    Farthing, M.2    Miller, C.3
  • 71
    • 33645862277 scopus 로고    scopus 로고
    • A local discontinuous Galerkin method for the KdV equation with boundary effect
    • H. Liu and J. Yan. A local discontinuous Galerkin method for the KdV equation with boundary effect. J. Comput. Phys., 215:197-218, 2006.
    • (2006) J. Comput. Phys , vol.215 , pp. 197-218
    • Liu, H.1    Yan, J.2
  • 72
    • 76449122240 scopus 로고    scopus 로고
    • The Direct Discontinuous Galerkin (DDG) methods for diffusion problems
    • H. Liu and J. Yan. The Direct Discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal., 47:675-698, 2008.
    • (2008) SIAM J. Numer. Anal , vol.47 , pp. 675-698
    • Liu, H.1    Yan, J.2
  • 73
    • 76449100234 scopus 로고    scopus 로고
    • Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation
    • T. Lu, W. Cai, and P. Zhang. Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation. Int. J. Numer. Anal. Model., 2:75-84, 2005.
    • (2005) Int. J. Numer. Anal. Model , vol.2 , pp. 75-84
    • Lu, T.1    Cai, W.2    Zhang, P.3
  • 74
    • 85128807957 scopus 로고    scopus 로고
    • Semi-implicit spectral deferred correction methods for ordinary differential equations
    • M. Minion. Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci., 1:471-500, 2003.
    • (2003) Commun. Math. Sci , vol.1 , pp. 471-500
    • Minion, M.1
  • 75
    • 0032209437 scopus 로고    scopus 로고
    • A discontinuous hp finite element method for diffusion problems
    • J. Oden, I. Babuvska, and C. Baumann. A discontinuous hp finite element method for diffusion problems. J. Comput. Phys., 146:491-519, 1998.
    • (1998) J. Comput. Phys , vol.146 , pp. 491-519
    • Oden, J.1    Babuvska, I.2    Baumann, C.3
  • 76
    • 0034206154 scopus 로고    scopus 로고
    • Adjoint recovery of superconvergent functionals fromPDE approximation
    • N. Pierce andM. Giles. Adjoint recovery of superconvergent functionals fromPDE approximation. SIAMReview, 42:247-264, 2000.
    • (2000) SIAMReview , vol.42 , pp. 247-264
    • Pierce andM, N.1    Giles2
  • 77
    • 0003630916 scopus 로고
    • Triangular mesh methods for the neutrontransport equation
    • Los Alamos Scientific Laboratory
    • W. Reed and T. Hill. Triangular mesh methods for the neutrontransport equation. La-ur-73-479, Los Alamos Scientific Laboratory, 1973.
    • (1973) La-ur-73-479
    • Reed, W.1    Hill, T.2
  • 78
    • 0037356074 scopus 로고    scopus 로고
    • J.-F. Remacle, J. Flaherty, andM. Shephard. An adaptive discontinuous Galerkin technique with an orthogonal basis applied to Rayleigh-Taylor ow instabilities. SIAM Rev., 45:53-72, 2003.
    • J.-F. Remacle, J. Flaherty, andM. Shephard. An adaptive discontinuous Galerkin technique with an orthogonal basis applied to Rayleigh-Taylor ow instabilities. SIAM Rev., 45:53-72, 2003.
  • 80
    • 12044253199 scopus 로고
    • Compacton: Solitons with finite wavelength
    • P. Rosenau and J. Hyman. Compacton: Solitons with finite wavelength. Phys. Rev. Lett., 70:564-567, 1993.
    • (1993) Phys. Rev. Lett , vol.70 , pp. 564-567
    • Rosenau, P.1    Hyman, J.2
  • 81
    • 76449107494 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for incompressible flow
    • D. Schotzau. Local discontinuous Galerkin methods for incompressible flow. Computational Fluid and Solid Mechanics, 1-2:1116-1118, 2003.
    • (2003) Computational Fluid and Solid Mechanics , vol.1-2 , pp. 1116-1118
    • Schotzau, D.1
  • 82
    • 84966251037 scopus 로고
    • TVB uniformly high-order schemes for conservation laws
    • C.-W. Shu. TVB uniformly high-order schemes for conservation laws. Math. Comp, 49:105-121, 1987.
    • (1987) Math. Comp , vol.49 , pp. 105-121
    • Shu, C.-W.1
  • 83
    • 0012894872 scopus 로고    scopus 로고
    • Different formulations of the discontinuous Galerkin method for the viscous terms
    • Z.-C. Shi, M. Mu,W. Xue, and J. Zou, editors, Science Press, Beijing
    • C.-W. Shu. Different formulations of the discontinuous Galerkin method for the viscous terms. In Z.-C. Shi, M. Mu,W. Xue, and J. Zou, editors, Advances in Scientific Computing, pages 144-155. Science Press, Beijing, 2001.
    • (2001) Advances in Scientific Computing , pp. 144-155
    • Shu, C.-W.1
  • 84
    • 67049120237 scopus 로고    scopus 로고
    • Discontinuous Galerkin methods: General approach and stability
    • S. Bertoluzza, S. Falletta, G. Russo, and C.-W. Shu, editors, Birkhäuser, Basel
    • C.-W. Shu. Discontinuous Galerkin methods: general approach and stability. In S. Bertoluzza, S. Falletta, G. Russo, and C.-W. Shu, editors, Numerical Solutions of Partial Differential Equations, pages 149-201. Birkhäuser, Basel, 2009.
    • (2009) Numerical Solutions of Partial Differential Equations , pp. 149-201
    • Shu, C.-W.1
  • 85
    • 45449125925 scopus 로고
    • Efficient implementation of essentially non-oscillatory shock-capturing schemes
    • C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77:439-471, 1988.
    • (1988) J. Comput. Phys , vol.77 , pp. 439-471
    • Shu, C.-W.1    Osher, S.2
  • 86
    • 0002885640 scopus 로고    scopus 로고
    • A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems
    • D. Kroner, M. Ohlberger, and C. Rhode, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, of, Springer
    • E. Süli. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In D. Kroner, M. Ohlberger, and C. Rhode, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, volume 5 of Lecture Notes in Computational Science and Engineering, pages 123-194. Springer, 1999.
    • (1999) Lecture Notes in Computational Science and Engineering , vol.5 , pp. 123-194
    • Süli, E.1
  • 87
    • 0001581452 scopus 로고    scopus 로고
    • Hp-DGFEM for partial differential equations with non-negative characteristic form
    • B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, of, Springer
    • E. Süli, C. Schwab, and P. Houston. Hp-DGFEM for partial differential equations with non-negative characteristic form. In B. Cockburn, G. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, volume 11 of Lecture Notes in Computational Science and Engineering, pages 221-230. Springer, 2000.
    • (2000) Lecture Notes in Computational Science and Engineering , vol.11 , pp. 221-230
    • Süli, E.1    Schwab, C.2    Houston, P.3
  • 89
    • 61349157975 scopus 로고    scopus 로고
    • Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion
    • M. van Raalte and B. van Leer. Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Commun. Comput. Phys., 5:683-693, 2009.
    • (2009) Commun. Comput. Phys , vol.5 , pp. 683-693
    • van Raalte, M.1    van Leer, B.2
  • 90
    • 41149086278 scopus 로고    scopus 로고
    • An adaptive strategy for the local discontinuous Galerkin method applied to porous media problems
    • E. Velazquez and P. Castillo. An adaptive strategy for the local discontinuous Galerkin method applied to porous media problems. Computer-Aided Civil and Infrastructure Engineering, 23:238-252, 2008.
    • (2008) Computer-Aided Civil and Infrastructure Engineering , vol.23 , pp. 238-252
    • Velazquez, E.1    Castillo, P.2
  • 91
    • 67349159773 scopus 로고    scopus 로고
    • TheWKB local discontinuous Galerkinmethod for the simulation of Schrödinger equation in a resonant tunneling diode
    • W.Wang and C.-W. Shu. TheWKB local discontinuous Galerkinmethod for the simulation of Schrödinger equation in a resonant tunneling diode. J. Sci. Comput., 40:360-374, 2009.
    • (2009) J. Sci. Comput , vol.40 , pp. 360-374
    • Wang, W.1    Shu, C.-W.2
  • 92
    • 35348961550 scopus 로고    scopus 로고
    • Efficient time discretization for local discontinuous Galerkin methods
    • Y. Xia, Y. Xu, and C.-W. Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete Contin. Dyn. Syst. Ser. B, 8:677-693, 2007.
    • (2007) Discrete Contin. Dyn. Syst. Ser. B , vol.8 , pp. 677-693
    • Xia, Y.1    Xu, Y.2    Shu, C.-W.3
  • 93
    • 35348877949 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for the Cahn-Hilliard type equations
    • Y. Xia, Y. Xu, and C.-W. Shu. Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys., 227:472-491, 2007.
    • (2007) J. Comput. Phys , vol.227 , pp. 472-491
    • Xia, Y.1    Xu, Y.2    Shu, C.-W.3
  • 94
    • 85150403697 scopus 로고    scopus 로고
    • Y. Xia, Y. Xu, and C.-W. Shu. Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. Commun. Comput. Phys., 5:821-835, 2009.
    • Y. Xia, Y. Xu, and C.-W. Shu. Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. Commun. Comput. Phys., 5:821-835, 2009.
  • 95
    • 65349107063 scopus 로고    scopus 로고
    • A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems
    • Z. Xie, Z. Zhang, and Z. Zhang. A numerical study of uniform superconvergence of LDG method for solving singularly perturbed problems. J. Comput. Math., 27:280-298, 2009.
    • (2009) J. Comput. Math , vol.27 , pp. 280-298
    • Xie, Z.1    Zhang, Z.2    Zhang, Z.3
  • 96
    • 3042791844 scopus 로고    scopus 로고
    • Local discontinuous galerkin methods for three classes of nonlinear wave equations
    • Y. Xu and C.-W. Shu. Local discontinuous galerkin methods for three classes of nonlinear wave equations. J. Comput. Math., 22:250-274, 2004.
    • (2004) J. Comput. Math , vol.22 , pp. 250-274
    • Xu, Y.1    Shu, C.-W.2
  • 97
    • 16844366209 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for nonlinear Schrödinger equations
    • Y. Xu and C.-W. Shu. Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys., 205:72-97, 2005.
    • (2005) J. Comput. Phys , vol.205 , pp. 72-97
    • Xu, Y.1    Shu, C.-W.2
  • 98
    • 22544467513 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations
    • Y. Xu and C.-W. Shu. Local discontinuous Galerkin methods for two classes of two dimensional nonlinear wave equations. Physica D, 208:21-58, 2005.
    • (2005) Physica D , vol.208 , pp. 21-58
    • Xu, Y.1    Shu, C.-W.2
  • 99
    • 33645906589 scopus 로고    scopus 로고
    • Local discontinuous Galerkinmethods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations
    • Y. Xu and C.-W. Shu. Local discontinuous Galerkinmethods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput.Methods Appl.Mech. Engrg., 195:3430-3447, 2006.
    • (2006) Comput.Methods Appl.Mech. Engrg , vol.195 , pp. 3430-3447
    • Xu, Y.1    Shu, C.-W.2
  • 100
    • 34447105456 scopus 로고    scopus 로고
    • Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations
    • Y. Xu and C.-W. Shu. Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations. Comput. Methods Appl. Mech. Engrg., 196:3805-3822, 2007.
    • (2007) Comput. Methods Appl. Mech. Engrg , vol.196 , pp. 3805-3822
    • Xu, Y.1    Shu, C.-W.2
  • 101
    • 55349125734 scopus 로고    scopus 로고
    • A local discontinuous Galerkin method for the Camassa-Holm equation. SIAMJ
    • Y. Xu and C.-W. Shu. A local discontinuous Galerkin method for the Camassa-Holm equation. SIAMJ. Numer. Anal., 46:1998-2021, 2008.
    • (2008) Numer. Anal , vol.46 , pp. 1998-2021
    • Xu, Y.1    Shu, C.-W.2
  • 102
    • 67649553021 scopus 로고    scopus 로고
    • Local discontinuous Galerkinmethod for theHunter-Saxton equation and its zero-viscosity and zero-dispersion limit. SIAMJ
    • Y. Xu and C.-W. Shu. Local discontinuous Galerkinmethod for theHunter-Saxton equation and its zero-viscosity and zero-dispersion limit. SIAMJ. Sci. Comput., 31:1249-1268, 2008.
    • (2008) Sci. Comput , vol.31 , pp. 1249-1268
    • Xu, Y.1    Shu, C.-W.2
  • 103
    • 67349111422 scopus 로고    scopus 로고
    • Local discontinuous Galerkin method for surface diffusion andWill-more flow of graphs
    • Y. Xu and C.-W. Shu. Local discontinuous Galerkin method for surface diffusion andWill-more flow of graphs. J. Sci. Comput., 40:375-390, 2009.
    • (2009) J. Sci. Comput , vol.40 , pp. 375-390
    • Xu, Y.1    Shu, C.-W.2
  • 104
    • 0038693389 scopus 로고    scopus 로고
    • J. Yan and C.-W. Shu. A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal., 40:769-791, 2002.
    • J. Yan and C.-W. Shu. A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal., 40:769-791, 2002.
  • 105
    • 1942475854 scopus 로고    scopus 로고
    • Local discontinuous Galerkin methods for partial differential equations with higher order derivatives
    • J. Yan and C.-W. Shu. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput., 17:27-47, 2002.
    • (2002) J. Sci. Comput , vol.17 , pp. 27-47
    • Yan, J.1    Shu, C.-W.2
  • 106
    • 33748546239 scopus 로고    scopus 로고
    • Discontinuous Galerkin method based on non-polynomial approximation spaces
    • L. Yuan and C.-W. Shu. Discontinuous Galerkin method based on non-polynomial approximation spaces. J. Comput. Phys., 218:295-323, 2006.
    • (2006) J. Comput. Phys , vol.218 , pp. 295-323
    • Yuan, L.1    Shu, C.-W.2
  • 107
    • 40749087210 scopus 로고    scopus 로고
    • Discontinuous Galerkin method for a class of elliptic multi-scale problems
    • L. Yuan and C.-W. Shu. Discontinuous Galerkin method for a class of elliptic multi-scale problems. Internat. J. Numer. Methods Fluids, 56:1017-1032, 2008.
    • (2008) Internat. J. Numer. Methods Fluids , vol.56 , pp. 1017-1032
    • Yuan, L.1    Shu, C.-W.2
  • 108
    • 0037357292 scopus 로고    scopus 로고
    • An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations
    • M. Zhang and C.-W. Shu. An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci., 13:395-413, 2003.
    • (2003) Math. Models Methods Appl. Sci , vol.13 , pp. 395-413
    • Zhang, M.1    Shu, C.-W.2
  • 109
    • 67349109198 scopus 로고    scopus 로고
    • Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method
    • Q. Zhang and Z. Wu. Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method. J. Sci. Comput., 38:127-148, 2009.
    • (2009) J. Sci. Comput , vol.38 , pp. 127-148
    • Zhang, Q.1    Wu, Z.2


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