A brief survey on discontinuous Galerkin methods in computational fluid dynamics

Chi-Wang Shu

doi:10.6052/1000-0992-13-059

Volume 43 Issue 6

Nov. 2013

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Chi-Wang Shu. A brief survey on discontinuous Galerkin methods in computational fluid dynamics[J]. Advances in Mechanics, 2013, 43(6): 541-554. doi: 10.6052/1000-0992-13-059

Citation:

Chi-Wang Shu. A brief survey on discontinuous Galerkin methods in computational fluid dynamics[J]. Advances in Mechanics, 2013, 43(6): 541-554. doi: 10.6052/1000-0992-13-059

Chi-Wang Shu. A brief survey on discontinuous Galerkin methods in computational fluid dynamics[J]. Advances in Mechanics, 2013, 43(6): 541-554. doi: 10.6052/1000-0992-13-059

Citation:

Chi-Wang Shu. A brief survey on discontinuous Galerkin methods in computational fluid dynamics[J]. Advances in Mechanics, 2013, 43(6): 541-554. doi: 10.6052/1000-0992-13-059

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A brief survey on discontinuous Galerkin methods in computational fluid dynamics

doi: 10.6052/1000-0992-13-059

Chi-Wang Shu

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Funds: The project was partially supported by US NSF grant DMS-1112700 and by the Open Fund of State Key Laboratory of High-temperature Gas Dynamics, China (No. 2011KF02).

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Corresponding author: Chi-Wang Shu
Received Date: 2013-08-27
Publish Date: 2013-11-25

Abstract

Abstract

Discontinuous Galerkin (DG) methods combine features in finite element methods(weak formulation, finite dimensional solution and test function spaces) and in finite volume methods (numerical fluxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In this article we will give a brief survey of DG methods, emphasizing their applications in computational fluid dynamics (CFD). We will discuss essential ingredients and properties of DG methods, and will also give a few examples of recent developments of DG methods which have facilitated their applications in CFD.
- discontinuous Galerkin methods,
- computational fluid dynamics

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References(77)

References

[1]	Arnold D N. 1982. An interior penalty finite elementmethod with discontinuous elements. SIAM Journal onNumerical Analysis, 39: 742-760.
[2]	Bassi F, Rebay S. 1997. A high-order accurate discontin-uous finite element method for the numerical solutionof the compressible Navier-Stokes equations. Journal ofComputational Physics, 131: 267-279.
[3]	Baumann C E, Oden J T. 1999. A discontinuous h-p fi-nite element method for convection-diffusion problems.Computer Methods in Applied Mechanics and Engineer-ing, 175: 311-341.
[4]	Biswas R, Devine K D, Flaherty J. 1994. Parallel, adaptivefinite element methods for conservation laws. Applied Numerical Mathematics, 14: 255-283.
[5]	Bokanowski O, Cheng Y, Shu C W. 2011. A discontinuous Galerkin solver for front propagation. SIAM Journal onScientific Computing, 33: 923-938.
[6]	Bokanowski O, Cheng Y, Shu C W. A discontinuous Galerkin scheme for front propagation with obsta-cles. Numerische Mathematik, to appear. DOI: 10.1007/s00211-013-0555-3.
[7]	Bokanowski O, Cheng Y, Shu C W. Convergence of dis-continuous Galerkin schemes for front propagation withobstacles. submitted to Mathematics of Computation.Burbeau A, Sagaut P, Bruneau Ch H. 2001. A problem-independent limiter for high-order Runge-Kutta discon-tinuous Galerkin methods. Journal of Computational Physics, 169: 111-150.
[8]	Canuto C, Fagnani F, Tilli P. 2012. An Eulerian approachto the analysis of Krause's consensus models. SIAM Journal on Control and Optimization, 50: 243-265.
[9]	Chen G Q, Liu H. 2003. Formation of ffi-shocks and vacuumstates in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM Journal on Mathematical Analysis, 34: 925-938.
[10]	Cheng Y, Shu C W. 2007. A discontinuous Galerkin finiteelement method for directly solving the Hamilton-Jacobiequations. Journal of Computational Physics, 223: 398-415.
[11]	Cheng Y, Shu C W. 2008. A discontinuous Galerkin finiteelement method for time dependent partial differentialequations with higher order derivatives. Mathematics ofComputation, 77: 699-730.
[12]	Cockburn B. 1999. Discontinuous Galerkin Methods for Convection-Dominated Problems. Berlin, Heidelberg,Springer 69-224.
[13]	Cockburn B, Hou S, Shu C W. 1990. The Runge-Kuttalocal projection discontinuous Galerkin finite elementmethod for conservation laws Ⅳ: the multidimensionalcase. Mathematics of Computation, 54: 545-581.
[14]	Cockburn B, Karniadakis G, Shu C W. 2000. The De-velopment of Discontinuous Galerkin Methods. Berlin,Heidelberg, Springer, 3-50.
[15]	Cockburn B, Lin S Y, Shu C W. 1989. TVB Runge-Kuttalocal projection discontinuous Galerkin finite elementmethod for conservation laws Ⅲ: one-dimensional sys-tems. Journal of Computational Physics, 84: 90-113.
[16]	Cockburn B, Shu C W. 1989. TVB Runge-Kutta local pro-jection discontinuous Galerkin finite element method forconservation laws Ⅱ: general framework. Mathematics of Computation, 52: 411-435.
[17]	Cockburn B, Shu C W. 1991. The Runge-Kutta local pro-jection P1-discontinuous-Galerkin finite element methodfor scalar conservation laws. Mathematical Modellingand Numerical Analysis (M2AN), 25: 337-361.
[18]	Cockburn B, Shu C W. 1998. The Runge-Kutta discontin-uous Galerkin method for conservation laws Ⅴ: multidi-mensional systems. Journal of Computational Physics,141: 199-224.
[19]	Cockburn B, Shu C W. 1998. The local discontinu-ous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Anal-ysis, 35: 2440-2463.
[20]	Cockburn B, Shu C W. 2001. Runge-Kutta discontinuousGalerkin methods for convection-dominated problems.Journal of Scientific Computing, 16: 173-261.
[21]	Cockburn B, Shu C W. 2005. Foreword for the special issueon discontinuous Galerkin method. Journal of Scientific Computing, 22-23: 1-3.
[22]	Cockburn B, Shu C W. 2009. Foreword for the special issueon discontinuous Galerkin method. Journal of Scientific Computing, 40: 1-3.
[23]	Dawson C. 2006. Foreword for the special issue on discon-tinuous Galerkin method. Computer Methods in Applied Mechanics and Engineering, 195: 3183.
[24]	Du J, Shu C W, Zhang M. 2013. A simple weighted essen-tially non-oscillatory limiter for the correction procedurevia reconstruction (CPR) framework. submitted to Ap-plied Numerical Mathematics.
[25]	Einfeldt B, Munz C D, Roe P L, Sjöogreen B. 1991. On Godunov-Type methods near low densities. Journal of Computational Physics, 92: 273-295.
[26]	Gottlieb S, Ketcheson D, Shu C W. 2011. Strong Stability Preserving Runge-Kutta and Multistep Time Discretiza-tions, Singapore: World Scientific.
[27]	Harten A. 1983. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics,49: 357-393.
[28]	Harten A, Lax P D, van Leer B. 1983. On upstream dif-ferencing and Godunov type schemes for hyperbolic con-servation laws. SIAM Review, 25: 35-61.
[29]	Hesthaven J, Warburton T. 2008. Nodal Discontinuous Galerkin Methods, New York: Springer.
[30]	Hou S, Liu X D. 2007. Solutions of multi-dimensional hy-perbolic systems of conservation laws by square entropycondition satisfying discontinuous Galerkin method.Journal of Scientific Computing, 31: 127-151.
[31]	Hu C, Shu C W. 1999. Weighted essentially non-oscillatorys chemes on triangular meshes. Journal of ComputationalPhysics, 150: 97-127.
[32]	Hu C, Shu C W. 1999. A discontinuous Galerkin finiteelement method for Hamilton-Jacobi equations. SIAMJournal on Scientific Computing, 21: 666-690.
[33]	Jiang G S, Shu C W. 1994. On cell entropy inequality fordiscontinuous Galerkin methods. Mathematics of Com-putation, 62: 531-538.
[34]	Jiang G S, Shu C W. 1996. Efficient implementationof weighted ENO schemes. Journal of ComputationalPhysics, 126: 202-228.
[35]	Kanschat G. 2007. Discontinuous Galerkin Methods for Viscous Flow, Wiesbaden Deutscher Universitätsverlag.Klockner A, Warburton T, Bridge J, Hesthaven J. 2010.Nodal discontinuous Galerkin methods on graphics pro-cessors. Journal of Computational Physics, 228: 7863-7882.
[36]	Krivodonova L, Xin J, Remacle J F, Chevaugeon N, Fla-herty J E. 2004. Shock detection and limiting with dis-continuous Galerkin methods for hyperbolic conservationlaws. Applied Numerical Mathematics, 48: 323-338.
[37]	LeVeque R J. 1990. Numerical Methods for Conservation Laws, Basel: Birkhauser Verlag.
[38]	Li B. 2006. Discontinuous Finite Elements in Fluid Dy-namics and Heat Transfer, Basel Birkhauser.
[39]	Li F, Shu C W. 2005. Reinterpretation and simplifiedimplementation of a discontinuous Galerkin method for Hamilton-Jacobi equations. Applied Mathematics Let-ters, 18: 1204-1209.
[40]	Liu H, Yan J. 2009. The direct discontinuous Galerkin(DDG) methods for diffusion problems. SIAM Journalon Numerical Analysis, 47: 675-698.
[41]	Liu H, Yan J. 2010. The direct discontinuous Galerkin(DDG) methods for diffusion with interface corrections.Communications in Computational Physics, 8: 541-564.
[42]	Liu X, Osher S, Chan T. 1994. Weighted essentially non-oscillatory schemes. Journal of Computational Physics,115: 200-212.
[43]	Oden J T, Babuvska I, Baumann C E. 1998. A discon-tinuous hp finite element method for diffusion problems.Journal of Computational Physics, 146: 491-519.
[44]	Perthame B. 1992. Second-order Boltzmann schemes forcompressible Euler equations in one and two space di-mensions. SIAM Journal on Numerical Analysis, 29:1-19.
[45]	Perthame B, Shu C W. 1996. On positivity preserving fi-nite volume schemes for Euler equations. NumerischeMathematik, 73: 119-130.
[46]	Qiu J M, Shu C W. 2011. Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoret-ical analysis and application to the Vlasov-Poisson sys-tem. Journal of Computational Physics, 230: 8386-8409.
[47]	Qiu J X, Shu C W. 2003. Hermite WENO schemes andtheir application as limiters for Runge-Kutta discontinu-ous Galerkin method: one-dimensional case. Journal ofComputational Physics, 193: 115-135.
[48]	Qiu J X, Shu C W. 2005. Hermite WENO schemes andtheir application as limiters for Runge-Kutta discontinu-ous Galerkin method Ⅱ: two dimensional case. Compu-ters and Fluids, 34: 642-663.
[49]	Qiu J X, Shu C W. 2005. Runge-Kutta discontinuousGalerkin method using WENO limiters. SIAM Journalon Scientific Computing, 26: 907-929.
[50]	Qiu J X, Shu C W. 2005. A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkinmethods using weighted essentially nonoscillatory lim-iters. SIAM Journal on Scientific Computing, 27: 995-1013.
[51]	Reed W H, Hill T R. 1973. Triangular mesh methods forthe neutron transport equation, Los Alamos ScientificLaboratory Report LA-UR-73-479, Los Alamos, NMRemacle J F, Flaherty J, Shephard M. 2003. An adaptivediscontinuous Galerkin technique with an orthogonal ba-sis applied to Rayleigh-Taylor flow instabilities. SIAM Review, 45: 53-72.
[52]	Riviére B. 2008. Discontinuous Galerkin Methods for Solv-ing Elliptic and Parabolic Equations: Theory and Imple-mentation, Philadelphia: SIAM
[53]	Rossmanith J A, Seal D C. 2011. A positivity-preserving high-order semi-Lagrangian discontinuousGalerkin scheme for the Vlasov-Poisson equations. Jour-nal of Computational Physics, 230: 6203-6232.
[54]	Shi J, Hu C, Shu C W. 2002. A technique of treating neg-ative weights in WENO schemes. Journal of Computa-tional Physics, 175: 108-127.
[55]	Shu C W. 1987. TVB uniformly high-order schemes for con-servation laws. Mathematics of Computation, 49: 105-121.
[56]	Shu C W. 2009. Discontinuous Galerkin methods: Generalapproach and stability. Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathemat-ics CRM Barcelona, 149-201.
[57]	Shu C W, Osher S. 1988. Efficient implementation of essen-tially non-oscillatory shock-capturing schemes. Journalof Computational Physics, 77: 439-471.
[58]	Toro E F. 1999. Riemann Solvers and Numerical Methodsfor Fluid Dynamics, Berlin: Springer-Verlag.
[59]	Wang Z J, Gao H. 2009. A unifying lifting colloca-tion penalty formulation including the discontinuousGalerkin, spectral volume/difference methods for conser-vation laws on mixed grids. Journal of Computational Physics, 228: 8161-8186.
[60]	Wheeler M F. 1978. An elliptic collocation-finite elementmethod with interior penalties. SIAM Journal on Nu-merical Analysis, 15: 152-161.
[61]	Xing Y, Zhang X, Shu C W. 2010. Positivity preservinghigh order well balanced discontinuous Galerkin meth-ods for the shallow water equations. Advances in Water Resources, 33: 1476-1493.
[62]	Xu Y, Shu C W. 2010. Local discontinuous Galerkin meth-ods for high-order time-dependent partial differentialequations. Communications in Computational Physics,7: 1-46.
[63]	Yan J, Osher S. 2011. A local discontinuous Galerkinmethod for directly solving Hamilton Jacobi equations.Journal of Computational Physics, 230: 232-244.
[64]	Yang Y, Shu C W. 2013. Discontinuous Galerkin method for hyperbolic equations involving ffi-singularities:negative-order norm error estimates and applications.Numerische Mathematik, 124: 753-781.
[65]	Yang Y, Wei D, Shu C W. 2013. Discontinuous Galerkinmethod for Krause's consensus models and pressureless Euler equations. Journal of Computational Physics, 252:109-127.
[66]	Zhang Q, Shu C W. 2010. Stability analysis and a priorierror estimates to the third order explicit Runge-Kuttadiscontinuous Galerkin method for scalar conservationlaws. SIAM Journal on Numerical Analysis, 48: 1038-1063.
[67]	Zhang X, Shu C W. 2010. On maximum-principle-satisfying high order schemes for scalar conservationlaws. Journal of Computational Physics, 229: 3091-3120.
[68]	Zhang X, Shu C W. 2010. On positivity preserving highorder discontinuous Galerkin schemes for compressibleEuler equations on rectangular meshes. Journal of Com-putational Physics, 229: 8918-8934.
[69]	Zhang X, Shu C W. 2011a. Positivity-preserving high orderdiscontinuous Galerkin schemes for compressible Eulerequations with source terms. Journal of ComputationalPhysics, 230: 1238-1248.
[70]	Zhang X, Shu C W. 2011b. Maximum-principle-satisfyingand positivity-preserving high order schemes for conser-vation laws: Survey and new developments, in: Proceed-ings of the Royal Society A, 467: 2752-2776.
[71]	Zhang X, Shu C W. 2012. A minimum entropy principleof high order schemes for gas dynamics equations. Nu-merische Mathematik, 121: 545-563.
[72]	Zhang X, Xia Y, Shu C W. 2012. Maximum-principle-satisfying and positivity-preserving high order discontin-uous Galerkin schemes for conservation laws on triangu-lar meshes. Journal of Scientific Computing, 50: 29-62.
[73]	Zhang Y, Zhang X, Shu C W. 2013. Maximum-principle-satisfying second order discontinuous Galerkin schemesfor convection-diffusion equations on triangular meshes.Journal of Computational Physics, 234: 295-316.
[74]	Zhang Y T, Shu C W. 2009. Third order WENO scheme onthree dimensional tetrahedral meshes. Communicationsin Computational Physics, 5: 836-848.
[75]	Zhong X, Shu C W. 2012. A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuousGalerkin methods. Journal of Computational Physics,232: 397-415.
[76]	Zhu J, Qiu J X, Shu C W, Dumbser M. 2008. Runge-Kutta discontinuous Galerkin method using WENO lim-iters II: Unstructured meshes. Journal of ComputationalPhysics, 227: 4330-4353.
[77]	Zhu J, Zhong X, Shu C W, Qiu J X. 2013. Runge-Kutta dis-continuous Galerkin method using a new type of WENOlimiters on unstructured meshes. Journal of Computa-tional Physics, 248: 200-220.