One-dimensional shock-capturing for high-order discontinuous Galerkin methods

SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted...

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Published in:	International journal for numerical methods in fluids Vol. 71; no. 6; pp. 737 - 755
Main Authors:	Casoni, E., Peraire, J., Huerta, A.
Format:	Journal Article Publication
Language:	English
Published:	Bognor Regis Blackwell Publishing Ltd 28.02.2013 Wiley Subscription Services, Inc
Subjects:	65 Numerical analysis 65E05 Numerical methods in complex analysis (potential theory, etc.) Algorithms Anàlisi numèrica Approximation artificial viscosity Classificació AMS Constraining Diffusion discontinuous Galerkin Elements finits, Mètode dels Finite volume method Galerkin methods high-order approximation Matemàtiques i estadística Mathematical analysis Mathematical models Mètodes numèrics Nonlinearity Numerical methods and algorithms Reproduction shock capturing Studies Àrees temàtiques de la UPC
ISSN:	0271-2091, 1097-0363
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Abstract	SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd. Figure 1 compares the artificial diffusion technique proposed here with high‐order limiters (described as moments’ in the figures), with the same number of degrees of freedom. The artificial diffusion technique outperforms the high‐order limiting scheme, demonstrating that it seems more beneficial to increase the degree of approximation than to refine the mesh in order to obtain accurate solutions.
AbstractList	Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd. SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high-order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright [copy 2012 John Wiley & Sons, Ltd. Figure 1 compares the artificial diffusion technique proposed here with high-order limiters (described as moments' in the figures), with the same number of degrees of freedom. The artificial diffusion technique outperforms the high-order limiting scheme, demonstrating that it seems more beneficial to increase the degree of approximation than to refine the mesh in order to obtain accurate solutions. Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high-order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Peer Reviewed SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high-order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high-order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright 2012 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT] SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd. Figure 1 compares the artificial diffusion technique proposed here with high‐order limiters (described as moments’ in the figures), with the same number of degrees of freedom. The artificial diffusion technique outperforms the high‐order limiting scheme, demonstrating that it seems more beneficial to increase the degree of approximation than to refine the mesh in order to obtain accurate solutions.
Author	Casoni, E. Huerta, A. Peraire, J.
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Cites_doi	10.1063/1.1699639 10.1016/0168-9274(94)90029-9 10.1006/jcph.1996.5572 10.1137/S1064827503425298 10.1137/0719052 10.1006/jcph.2001.6718 10.1016/0021-9991(89)90183-6 10.1007/978-0-387-72067-8 10.1023/A:1012873910884 10.2307/2008474 10.1016/0021-9991(88)90177-5 10.1016/j.jcp.2009.11.010 10.1017/CBO9780511791253 10.1007/978-1-4020-9231-2_21 10.2514/6.2006-112 10.1016/j.jcp.2007.05.011 10.1137/S0036142997316712 10.1137/0721016 10.1007/978-3-0348-8629-1 10.1016/S0377-0427(00)00512-4 10.1137/S003614450036757X 10.1002/0470013826
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References	Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing 2001; 16(3):173-261. Hesthaven JS, Warburton T. Nodal Discontinuous Galerkin Methods, Texts in Applied Mathematics, Vol. 54, Springer: New York, 2008. Algorithms, analysis, and applications. Donea J, Huerta A. Finite Element Methods for Flow Problems. John Wiley & Sons: Chichester, 2003. von Neumann J, Richtmyer RD. A method for the numerical calculation of hydrodynamic shocks. Journal of Applied Physics 1950; 21:232-237. LeVeque RJ. Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press: Cambridge, 2002. Krivodonova L. Limiters for high-order discontinuous Galerkin methods. Journal of Computational Physics 2007; 226(1):879-896. Bassi F, Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics 1997; 131(2):267-279. Biswas R, Devine KD, Flaherty JE. Parallel, adaptive finite element methods for conservation laws. Applied Numerical Mathematics 1994; 14(1-3):255-283. Cockburn B, Lin SY, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. Journal of Computational Physics 1989; 84(1):90-113. Shu CW. Total-variation-diminishing time discretizations. Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing 1988; 9(6):1073-1084. Bassi F, Rebay S. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. International Journal for Numerical Methods in Engineering 2001; 40(10):197-207. Cockburn B, Shu C-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis 1998; 35(6):2440-2463. Cockburn B, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of Computation 1989; 52(186):411-435. Qiu J, Shu CW. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM Journal on Scientific Computing 2005; 26(3):907-929. Barter GE, Darmofal DL. Shock capturing with PDE-based artificial viscosity for DGFEM: part I. formulation. Journal of Computational Physics 2010; 229(5):1810-1827. Cockburn B. Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. Journal of Computational and Applied Mathematics 2001; 128(1-2):187-204. Shu CW, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. Journal of Computational Physics 1988; 77(2):439-471. Folland GB. Fourier Analysis and its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software: Pacific Grove, CA, 1992. Huerta A, Casoni E, Peraire J. A simple shock-capturing technique for high-order discontinuous Galerkin methods. International Journal for Numerical Methods in Fluids 2011. Available from: http:/dx.doi.org/10.1002/fld.2654. Gottlieb S, Shu CW, Tadmor E. Strong stability-preserving high-order time discretization methods. SIAM Review 2001; 43(1):89-112. Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 1982; 19(4):742-760. Peraire J, Persson P-O. The compact discontinuous Galerkin (CDG) method for elliptic problems. Society for Industrial and Applied Mathematics. Journal on Scientific Computing 1988; 30(4):1806-1824. LeVeque RJ. Numerical Methods for Conservation Laws (2nd edn), Lectures in Mathematics ETH Zürich, Birkhäuser Verlag: Basel, 1992. Burbeau A, Sagaut P, Bruneau CH. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational Physics 2001; 169(1):111-150. Osher S. Riemann solvers, the entropy condition, and difference approximations. SIAM Journal on Numerical Analysis 1984; 21(2):217-235. Casoni E, Peraire J, Huerta A. Un método de captura de choques basado en las funciones de forma para galerkin discontinuo en alto orden. Revista Internacional de Métodos Numéricos en Ingeniería 2011. To appear. 1989; 84 2007; 226 1982; 19 2011 2010; 229 1950; 21 1997; 131 1984; 21 1988; 77 2008; 54 1992 1988; 30 2003 2006; 2 2002 2005; 26 2001; 40 2001; 169 2001; 128 2001; 43 1989; 52 2009; 14 1988; 9 1994; 14 2001; 16 1998; 35 e_1_2_8_28_1 e_1_2_8_29_1 Huerta A (e_1_2_8_12_1) 2011 e_1_2_8_24_1 e_1_2_8_25_1 Peraire J (e_1_2_8_22_1) 1988; 30 Casoni E (e_1_2_8_26_1) 2011 e_1_2_8_3_1 e_1_2_8_2_1 e_1_2_8_5_1 e_1_2_8_7_1 e_1_2_8_6_1 e_1_2_8_9_1 e_1_2_8_8_1 e_1_2_8_21_1 e_1_2_8_23_1 Folland GB (e_1_2_8_27_1) 1992 Bassi F (e_1_2_8_20_1) 2001; 40 e_1_2_8_17_1 e_1_2_8_18_1 e_1_2_8_19_1 e_1_2_8_13_1 e_1_2_8_14_1 e_1_2_8_15_1 e_1_2_8_16_1 Shu CW (e_1_2_8_4_1) 1988; 9 e_1_2_8_10_1 e_1_2_8_11_1
References_xml	– reference: Casoni E, Peraire J, Huerta A. Un método de captura de choques basado en las funciones de forma para galerkin discontinuo en alto orden. Revista Internacional de Métodos Numéricos en Ingeniería 2011. To appear. – reference: LeVeque RJ. Numerical Methods for Conservation Laws (2nd edn), Lectures in Mathematics ETH Zürich, Birkhäuser Verlag: Basel, 1992. – reference: Qiu J, Shu CW. Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM Journal on Scientific Computing 2005; 26(3):907-929. – reference: Donea J, Huerta A. Finite Element Methods for Flow Problems. John Wiley & Sons: Chichester, 2003. – reference: Cockburn B, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Mathematics of Computation 1989; 52(186):411-435. – reference: Shu CW, Osher S. Efficient implementation of essentially nonoscillatory shock-capturing schemes. Journal of Computational Physics 1988; 77(2):439-471. – reference: Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing 2001; 16(3):173-261. – reference: Cockburn B. Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. Journal of Computational and Applied Mathematics 2001; 128(1-2):187-204. – reference: Bassi F, Rebay S. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. International Journal for Numerical Methods in Engineering 2001; 40(10):197-207. – reference: Burbeau A, Sagaut P, Bruneau CH. A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational Physics 2001; 169(1):111-150. – reference: Biswas R, Devine KD, Flaherty JE. Parallel, adaptive finite element methods for conservation laws. Applied Numerical Mathematics 1994; 14(1-3):255-283. – reference: Osher S. Riemann solvers, the entropy condition, and difference approximations. SIAM Journal on Numerical Analysis 1984; 21(2):217-235. – reference: Cockburn B, Shu C-W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM Journal on Numerical Analysis 1998; 35(6):2440-2463. – reference: Cockburn B, Lin SY, Shu CW. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. Journal of Computational Physics 1989; 84(1):90-113. – reference: Krivodonova L. Limiters for high-order discontinuous Galerkin methods. Journal of Computational Physics 2007; 226(1):879-896. – reference: Bassi F, Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics 1997; 131(2):267-279. – reference: Gottlieb S, Shu CW, Tadmor E. Strong stability-preserving high-order time discretization methods. SIAM Review 2001; 43(1):89-112. – reference: Hesthaven JS, Warburton T. Nodal Discontinuous Galerkin Methods, Texts in Applied Mathematics, Vol. 54, Springer: New York, 2008. Algorithms, analysis, and applications. – reference: LeVeque RJ. Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press: Cambridge, 2002. – reference: von Neumann J, Richtmyer RD. A method for the numerical calculation of hydrodynamic shocks. Journal of Applied Physics 1950; 21:232-237. – reference: Huerta A, Casoni E, Peraire J. A simple shock-capturing technique for high-order discontinuous Galerkin methods. International Journal for Numerical Methods in Fluids 2011. Available from: http:/dx.doi.org/10.1002/fld.2654. – reference: Arnold DN. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 1982; 19(4):742-760. – reference: Peraire J, Persson P-O. 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Snippet	SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent... Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a... SUMMARY Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent...
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SubjectTerms	65 Numerical analysis 65E05 Numerical methods in complex analysis (potential theory, etc.) Algorithms Anàlisi numèrica Approximation artificial viscosity Classificació AMS Constraining Diffusion discontinuous Galerkin Elements finits, Mètode dels Finite volume method Galerkin methods high-order approximation Matemàtiques i estadística Mathematical analysis Mathematical models Mètodes numèrics Nonlinearity Numerical methods and algorithms Reproduction shock capturing Studies Àrees temàtiques de la UPC
Title	One-dimensional shock-capturing for high-order discontinuous Galerkin methods
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