nach oben

Journal of Scientific Computing

Erschienen in:

01.02.2020

The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws

verfasst von: Shuai Lou, Chao Yan, Li-Bin Ma, Zhen-Hua Jiang

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, we develop a Lax–Wendroff type time discretization method for high order Flux Reconstruction scheme to solve hyperbolic conservation laws. Through Cauchy–Kowalewski procedure, the resulting Lax–Wendroff Flux Reconstruction (LWFR) scheme is an alternative spatial–temporal coupling method to the popular Runge–Kutta Flux Reconstruction (RKFR) scheme. LWFR is a one-step explicit high order discontinuous finite element method and its discretization procedure is more compact and effective for certain problems than that of RKFR. Furthermore, aiming at accurate simulation of discontinuity, we propose a robust local artificial viscosity formulation of LWFR for the first time. A collection of successful numerical experiments show that LWFR can give essentially non-oscillatory and sharp solutions for discontinuity, and maintain designed order of accuracy for smooth regions, both in one-dimensional and two-dimensional Euler equations. In conclusion, LWFR scheme is cost effective and accuracy-preserving for certain problems and can be an attractive candidate to solve hyperbolic conservation laws.

Vorheriger Artikel Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Nächster Artikel Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts Discretizations in Curvilinear Coordinates

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

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!

Jetzt informieren

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!

Jetzt informieren

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!

Jetzt informieren

Nur mit Berechtigung zugänglich

Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference, AIAA 2007-4079 (2007)

Wang, Z.J., Gao, H.Y.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228, 8161–8186 (2009)MathSciNetMATH

Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47, 50–72 (2010)MathSciNetMATH

Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)MathSciNetMATH

Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)MathSciNetMATH

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

Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids. J. Comput. Phys. 181, 186–221 (2002)MathSciNetMATH

Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244–261 (1996)MathSciNetMATH

Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13, 217–237 (1960)MATH

10.

Jameson, A., Schmidt, W., Turkel, E.L.I.: Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes. In: 14th Fluid and Plasma Dynamics Conference, AIAA 1981-1259 (1981)

11.

Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge–Kutta schemes. In: Technical Report TM 109112 NASA Langley Research Center (1994)

12.

Gottlieb, S., Grant, Z., Higgs, D.: Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order. Math. Comput. 84, 2743–2761 (2015)MathSciNetMATH

13.

Huynh, H.T.: A reconstruction approach to high-order schemnes including discontinuous Galerkin for diffusion. In: 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, AIAA 2009-0403 (2009)

14.

Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45, 348–358 (2010)MathSciNetMATH

15.

Castonguay, P., Willians, D.M., Vincent, P.E., Jameson, A.: Energy stable flux reconstruction schemes for advection–diffusion problems. Comput. Methods Appl. Mech. Eng. 267, 400–417 (2013)MathSciNetMATH

16.

Castonguay, P., Vincent, P.E., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes for triangular elements. J. Sci. Comput. 51, 224–256 (2011)MathSciNetMATH

17.

Williams, D.M., Castonguay, P., Vincent, P.E., Jameson, A.: Energy stable flux reconstruction schemes for advection–diffusion problems on triangles. J. Comput. Phys. 250, 53–76 (2013)MathSciNetMATH

18.

Sheshadri, A., Jameson, A.: On the stability of the flux reconstruction schemes on quadrilateral elements for the linear advection equation. J. Sci. Comput. 67, 769–790 (2015)MathSciNetMATH

19.

Vincent, P.E., Castonguay, P., Jameson, A.: Insights from von Neumann analysis of high-order flux reconstruction schemes. J. Comput. Phys. 230, 8134–8154 (2011)MathSciNetMATH

20.

Asthana, K., Jameson, A.: High-order flux reconstruction schemes with minimal dispersion and dissipation. J. Sci. Comput. 62, 913–944 (2015)MathSciNetMATH

21.

Vermeire, B.C., Vincent, P.E.: On the behaviour of fully-discrete flux reconstruction schemes. Comput. Methods Appl. Mech. Eng. 315, 1053–1079 (2017)MathSciNet

22.

Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016)MathSciNetMATH

23.

Alhawwary, M., Wang, Z.J.: Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws. J. Comput. Phys. 373, 835–862 (2018)MathSciNetMATH

24.

Crean, J., Hicken, J.E., Fernández, D.C.D.R., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. 356, 410–438 (2018)MathSciNetMATH

25.

Ranocha, H., Glaubitz, J., Öffner, P., Sonar, T.: Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators. Appl. Numer. Math. 128, 1–23 (2018)MathSciNetMATH

26.

Vermeire, B.C., Vincent, P.E.: On the properties of energy stable flux reconstruction schemes for implicit large Eddy simulation. J. Comput. Phys. 327, 368–388 (2016)MathSciNetMATH

27.

Qiu, J.X., Shu, C.W.: Finite difference WENO schemes with Lax–Wendroff-type time discretizations. SIAM J. Sci. Comput. 24, 2185–2198 (2003)MathSciNetMATH

28.

Jiang, Y., Shu, C.W., Zhang, M.P.: An alternative formulation of finite difference weighted ENO schemes with Lax–Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35, A1137–A1160 (2013)MathSciNetMATH

29.

Zorío, D., Baeza, A., Mulet, P.: An approximate Lax–Wendroff-type procedure for high order accurate schemes for hyperbolic conservation laws. J. Sci. Comput. 71, 246–273 (2016)MathSciNetMATH

30.

Qiu, J.X., Dumbser, M., Shu, C.W.: The discontinuous Galerkin method with Lax–Wendroff type time discretizations. Comput. Methods Appl. Mech. Eng. 194, 4528–4543 (2005)MathSciNetMATH

31.

Guo, W., Qiu, J.M., Qiu, J.X.: A New Lax–Wendroff discontinuous Galerkin method with superconvergence. J. Sci. Comput. 65, 299–326 (2014)MathSciNetMATH

32.

Bürger, R., Kenettinkara, S.K., Zorío, D.: Approximate Lax–Wendroff discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Math Appl. 74, 1288–1310 (2017)MathSciNetMATH

33.

Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 609–618 (2002)MathSciNetMATH

34.

Mani, A., Larsson, J., Moin, P.: Suitability of artificial bulk viscosity for large-eddy simulation of turbulent flows with shocks. J. Comput. Phys. 228, 7368–7374 (2009)MATH

35.

Haga, T., Kawai, S.: On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method. J. Comput. Phys. 376, 534–563 (2019)MathSciNetMATH

36.

Fiorina, B., Lele, S.K.: An artificial nonlinear diffusivity method for supersonic reacting flows with shocks. J. Comput. Phys. 222, 246–264 (2007)MathSciNetMATH

37.

Deng, X., Jiang, Z.H., Xiao, F., Yan, C.: Implicit large eddy simulation of compressible turbulence flow with PnTm–BVD scheme. Appl. Math. Model. 77, 17–31 (2020)MathSciNet

38.

Shu, C.W.: TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105–121 (1987)MathSciNetMATH

39.

Qiu, J.X., Shu, C.W.: Runge–Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26, 907–929 (2005)MathSciNetMATH

40.

Zhong, X.H., Shu, C.W.: A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods. J. Comput. Phys. 232, 397–415 (2013)MathSciNet

41.

Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids Basic formulation. J. Comput. Phys. 178, 210–251 (2002)MathSciNetMATH

42.

Abeele, K.V.D., Lacor, C., Wang, Z.J.: On the connection between the spectral volume and the spectral difference method. J. Comput. Phys. 227, 877–885 (2007)MathSciNetMATH

43.

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)MathSciNetMATH

44.

Chen, S.S., Yan, C., Lin, B.X., Liu, L.Y., Yu, J.: Affordable shock-stable item for Godunov-type schemes against carbuncle phenomenon. J. Comput. Phys. 373, 662–672 (2018)MathSciNetMATH

45.

Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978)MathSciNetMATH

46.

Lax, P.D.: Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7, 159–193 (1954)MathSciNetMATH

47.

Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009)MATH

48.

Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78 (1989)MathSciNetMATH

49.

Zhang, T., Zheng, Y.X.: Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21, 593–630 (1990)MathSciNetMATH

50.

Lax, P.D., Liu, X.D.: Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19, 319–340 (1998)MathSciNetMATH

51.

Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-WENO finite difference scheme with conjugate fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38, 691–711 (2016)MathSciNetMATH

52.

Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)MathSciNetMATH

53.

Abbassi, H., Mashayek, F., Jacobs, G.B.: Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method. Comput. Fluids 98, 152–163 (2014)MathSciNetMATH

Titel: The Flux Reconstruction Method with Lax–Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws
verfasst von: Shuai Lou
Chao Yan
Li-Bin Ma
Zhen-Hua Jiang
Publikationsdatum: 01.02.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01146-8

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 2/2020

Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

No-Collision Transportation Maps

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties