## Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: Analysis and application in one dimension

HTML articles powered by AMS MathViewer

- by Christian Klingenberg, Gero Schnücke and Yinhua Xia PDF
- Math. Comp.
**86**(2017), 1203-1232 Request permission

## Abstract:

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, $\mathrm {L}^{2}$ stability and error estimates are proven. More precisely, we prove the sub-optimal ($k+\frac {1}{2}$) convergence for monotone fluxes and optimal ($k+1$) convergence for an upwind flux when a piecewise $P^k$ polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover, we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.## References

- Philippe G. Ciarlet,
*The finite element method for elliptic problems*, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR**1930132**, DOI 10.1137/1.9780898719208 - Bernardo Cockburn,
*Discontinuous Galerkin methods for convection-dominated problems*, High-order methods for computational physics, Lect. Notes Comput. Sci. Eng., vol. 9, Springer, Berlin, 1999, pp. 69–224. MR**1712278**, DOI 10.1007/978-3-662-03882-6_{2} - Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu,
*The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case*, Math. Comp.**54**(1990), no. 190, 545–581. MR**1010597**, DOI 10.1090/S0025-5718-1990-1010597-0 - Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu,
*TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems*, J. Comput. Phys.**84**(1989), no. 1, 90–113. MR**1015355**, DOI 10.1016/0021-9991(89)90183-6 - Bernardo Cockburn and Chi-Wang Shu,
*TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework*, Math. Comp.**52**(1989), no. 186, 411–435. MR**983311**, DOI 10.1090/S0025-5718-1989-0983311-4 - Bernardo Cockburn and Chi-Wang Shu,
*Runge-Kutta discontinuous Galerkin methods for convection-dominated problems*, J. Sci. Comput.**16**(2001), no. 3, 173–261. MR**1873283**, DOI 10.1023/A:1012873910884 - J. Donea, A. Huerta, J.-P. Ponthot, and A. Rodríguez-Ferran,
*Arbitrary Lagrangian-Eulerian methods*, in Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Erwin Stein, Rene De Borst and Thomas J.R. Hughes (Eds.), Wiley, 2004. - R. J. DiPerna,
*Convergence of approximate solutions to conservation laws*, Arch. Rational Mech. Anal.**82**(1983), no. 1, 27–70. MR**684413**, DOI 10.1007/BF00251724 - Charbel Farhat, Philippe Geuzaine, and Céline Grandmont,
*The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids*, J. Comput. Phys.**174**(2001), no. 2, 669–694. MR**1868101**, DOI 10.1006/jcph.2001.6932 - R. Fazio and R. J. LeVeque,
*Moving-mesh methods for one-dimensional hyperbolic problems using CLAWPACK*, Comput. Math. Appl.**45**(2003), no. 1-3, 273–298. Numerical methods in physics, chemistry, and engineering. MR**1991370**, DOI 10.1016/S0898-1221(03)80019-6 - Sigal Gottlieb and Chi-Wang Shu,
*Total variation diminishing Runge-Kutta schemes*, Math. Comp.**67**(1998), no. 221, 73–85. MR**1443118**, DOI 10.1090/S0025-5718-98-00913-2 - Hervé Guillard and Charbel Farhat,
*On the significance of the geometric conservation law for flow computations on moving meshes*, Comput. Methods Appl. Mech. Engrg.**190**(2000), no. 11-12, 1467–1482. MR**1807009**, DOI 10.1016/S0045-7825(00)00173-0 - Ami Harten,
*High resolution schemes for hyperbolic conservation laws*, J. Comput. Phys.**49**(1983), no. 3, 357–393. MR**701178**, DOI 10.1016/0021-9991(83)90136-5 - U. W. Hochstrasser,
*Orthogonal polynomials*, in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz and I. Stegun, eds.), Applied Mathematics Series, vol. 55, National Bureau of Standards, Washington, D.C., 1964, pp. 771–792. - Guang Shan Jiang and Chi-Wang Shu,
*On a cell entropy inequality for discontinuous Galerkin methods*, Math. Comp.**62**(1994), no. 206, 531–538. MR**1223232**, DOI 10.1090/S0025-5718-1994-1223232-7 - C. Johnson and J. Pitkäranta,
*An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation*, Math. Comp.**46**(1986), no. 173, 1–26. MR**815828**, DOI 10.1090/S0025-5718-1986-0815828-4 - D. Kereš, M. Vogelsberger, D. Sijacki, V. Springel, and L. Hernquist,
*Moving-mesh cosmology: Characteristics of galaxies and haloes*, Oxford J. Sci. & Math. MNRAS**425**(2012), 2027–2048. - S. N. Kružkov,
*First order quasilinear equations with several independent variables.*, Mat. Sb. (N.S.)**81 (123)**(1970), 228–255 (Russian). MR**0267257** - P. Lasaint and P.-A. Raviart,
*On a finite element method for solving the neutron transport equation*, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR**0658142** - M. Lesoinne and C. Farhat,
*Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations*, Comput. Methods Appl. Mech. Engrg.**134**(1996), 71–90. - P. D. Thomas and C. K. Lombard,
*Geometric conservation law and its application to flow computations on moving grids*, AIAA J.**17**(1979), no. 10, 1030–1037. MR**544850**, DOI 10.2514/3.61273 - I. Lomtev, R. M. Kirby, and G. E. Karniadakis,
*A discontinuous Galerkin ALE method for compressible viscous flows in moving domains*, J. Comput. Phys.**155**(1999), no. 1, 128–159. MR**1716493**, DOI 10.1006/jcph.1999.6331 - V.-T. Nguyen,
*An arbitrary Lagrangian-Eulerian discontinuous Galerkin method for simulations of flows over variable geometries*, J. Fluids and Struc.**26**(2010), 312–329. - P.-O. Persson, J. Bonet, and J. Peraire,
*Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains*, Comput. Methods Appl. Mech. Engrg.**198**(2009), 1585–1595. - Todd E. Peterson,
*A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation*, SIAM J. Numer. Anal.**28**(1991), no. 1, 133–140. MR**1083327**, DOI 10.1137/0728006 - B. A. Robinson, H. T. Y. Yang, and J. T. Batina,
*Aeroelastic analysis of wings using the Euler equations with a deforming mesh*, J. Aircraft**28**(1991), 781–788. - Chi-Wang Shu,
*TVB uniformly high-order schemes for conservation laws*, Math. Comp.**49**(1987), no. 179, 105–121. MR**890256**, DOI 10.1090/S0025-5718-1987-0890256-5 - Qiang Zhang and Chi-Wang Shu,
*Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws*, SIAM J. Numer. Anal.**42**(2004), no. 2, 641–666. MR**2084230**, DOI 10.1137/S0036142902404182 - Qiang Zhang and Chi-Wang Shu,
*Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws*, SIAM J. Numer. Anal.**44**(2006), no. 4, 1703–1720. MR**2257123**, DOI 10.1137/040620382 - Qiang Zhang and Chi-Wang Shu,
*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**(2010), no. 3, 1038–1063. MR**2669400**, DOI 10.1137/090771363 - Xiangxiong Zhang and Chi-Wang Shu,
*On maximum-principle-satisfying high order schemes for scalar conservation laws*, J. Comput. Phys.**229**(2010), no. 9, 3091–3120. MR**2601091**, DOI 10.1016/j.jcp.2009.12.030

## Additional Information

**Christian Klingenberg**- Affiliation: University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
- MR Author ID: 221691
- Email: klingen@mathematik.uni-wuerzburg.de
**Gero Schnücke**- Affiliation: University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
- Email: gero.schnuecke@web.de
**Yinhua Xia**- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
- MR Author ID: 819188
- Email: yhxia@ustc.edu.cn
- Received by editor(s): June 5, 2015
- Received by editor(s) in revised form: September 25, 2015
- Published electronically: June 20, 2016
- Additional Notes: The third author is the corresponding author. The research of the third author was supported by NSFC grants No. 11371342 and No. 11471306.
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1203-1232 - MSC (2010): Primary 65M12, 65M15, 65M60; Secondary 35L65
- DOI: https://doi.org/10.1090/mcom/3126
- MathSciNet review: 3614016