## Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws on moving simplex meshes

HTML articles powered by AMS MathViewer

- by
Pei Fu, Gero Schnücke and Yinhua Xia
**HTML**| PDF - Math. Comp.
**88**(2019), 2221-2255 Request permission

## Abstract:

In Klingenberg, Schnücke, and Xia (Math. Comp. 86 (2017), 1203–1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semidiscrete method the $\mathrm {L}^2$-stability will be proven. Furthermore, an error estimate which provides the suboptimal ($k+\frac {1}{2}$) convergence with respect to the $\mathrm {L}^{\infty }\!\left (0,T;\mathrm {L}^{2}\!\left (\Omega \right )\right )$-norm will be presented when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree $k$. The two-dimensional fully-discrete explicit method will be combined with the bound-preserving limiter developed by Zhang, Xia, and Shu (in J. Sci. Comput. 50 (2012), 29–62). This limiter does not affect the high-order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter, the validity of a discrete maximum principle will be proven. The numerical stability, robustness, and accuracy of the method will be shown by a variety of two-dimensional computational experiments on moving triangular meshes.## References

- R. Bellman.
*Introduction to matrix analysis, volume 19 of Classics in Applied Mathematics*, Classics in Applied Mathematics (SIAM, Philadelphia, PA, 2nd edition, 1987. - Walter Boscheri and Michael Dumbser,
*Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes*, J. Comput. Phys.**346**(2017), 449–479. MR**3670647**, DOI 10.1016/j.jcp.2017.06.022 - Philippe G. Ciarlet,
*Linear and nonlinear functional analysis with applications*, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. MR**3136903** - 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 and Chi-Wang Shu,
*The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems*, J. Comput. Phys.**141**(1998), no. 2, 199–224. MR**1619652**, DOI 10.1006/jcph.1998.5892 - 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: Part 1. Fundamentals*, in Encyclopedia of Computational Mechanics, E. Stein, R. De Borst and T. J. R. Hughes (eds.), Chapter 14, Wiley, 2004. - S. Étienne, A. Garon, and D. Pelletier,
*Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow*, J. Comput. Phys.**228**(2009), no. 7, 2313–2333. MR**2501688**, DOI 10.1016/j.jcp.2008.11.032 - 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 - 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 - Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor,
*Strong stability-preserving high-order time discretization methods*, SIAM Rev.**43**(2001), no. 1, 89–112. MR**1854647**, DOI 10.1137/S003614450036757X - 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 - 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 - Christian Klingenberg, Gero Schnücke, and Yinhua Xia,
*Arbitrary Lagrangian-Eulerian discontinuous Galerkin method for conservation laws: analysis and application in one dimension*, Math. Comp.**86**(2017), no. 305, 1203–1232. MR**3614016**, DOI 10.1090/mcom/3126 - Christian Klingenberg, Gero Schnücke, and Yinhua Xia,
*An arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for Hamilton-Jacobi equations*, J. Sci. Comput.**73**(2017), no. 2-3, 906–942. MR**3719613**, DOI 10.1007/s10915-017-0471-2 - David A. Kopriva,
*Metric identities and the discontinuous spectral element method on curvilinear meshes*, J. Sci. Comput.**26**(2006), no. 3, 301–327. MR**2226533**, DOI 10.1007/s10915-005-9070-8 - David A. Kopriva, Andrew R. Winters, Marvin Bohm, and Gregor J. Gassner,
*A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes*, Comput. & Fluids**139**(2016), 148–160. MR**3556311**, DOI 10.1016/j.compfluid.2016.05.023 - S. N. Kružkov,
*First order quasilinear equations in several independent variables*, Math. USSR-Sbornik**10**(1970), 217–243. - D. Kuzmin,
*A guide to numerical methods for transport equations*, Lecture, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2010. - D. Kuzmin and S. Turek,
*Flux correction tools for finite elements*, J. Comput. Phys.**175**(2002), no. 2, 525–558. MR**1880117**, DOI 10.1006/jcph.2001.6955 - 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 - Juan Luo, Chi-Wang Shu, and Qiang Zhang,
, ESAIM Math. Model. Numer. Anal.*A priori*error estimates to smooth solutions of the third order Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws**49**(2015), no. 4, 991–1018. MR**3371901**, DOI 10.1051/m2an/2014063 - Dimitri J. Mavriplis and Zhi Yang,
*Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes*, J. Comput. Phys.**213**(2006), no. 2, 557–573. MR**2207250**, DOI 10.1016/j.jcp.2005.08.018 - Cesar A. Acosta Minoli and David A. Kopriva,
*Discontinuous Galerkin spectral element approximations on moving meshes*, J. Comput. Phys.**230**(2011), no. 5, 1876–1902. MR**2764015**, DOI 10.1016/j.jcp.2010.11.038 - V. T. Nguyen,
*An arbitrary Lagrangian-Eulerian discontinuous Galerkin method for simulations of flows over variable geometries*, J. Fluid. Struct.**26**(2010), 312–329. - Stanley Osher,
*Convergence of generalized MUSCL schemes*, SIAM J. Numer. Anal.**22**(1985), no. 5, 947–961. MR**799122**, DOI 10.1137/0722057 - 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. - W. H. Reed and T. R. Hill,
*Triangular mesh method for the neutron transport equation*, Technical report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973. - B. A. Robinson, H. T. 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,
*Total-variation-diminishing time discretizations*, SIAM J. Sci. Statist. Comput.**9**(1988), no. 6, 1073–1084. MR**963855**, DOI 10.1137/0909073 - Raymond J. Spiteri and Steven J. Ruuth,
*A new class of optimal high-order strong-stability-preserving time discretization methods*, SIAM J. Numer. Anal.**40**(2002), no. 2, 469–491. MR**1921666**, DOI 10.1137/S0036142901389025 - L. Wang and P. O. Persson,
*High-order discontinuous Galerkin simulations on moving domains using ALE formulations and local remeshing and projections*, 53rd AIAA Aerospace Sciences Meeting, AIAA SciTech Forum (AIAA 2015-0820). Available via http://dx.doi.org/10.2514/6.2015-0820 - Yan Xu and Chi-Wang Shu,
*Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection-diffusion and KdV equations*, Comput. Methods Appl. Mech. Engrg.**196**(2007), no. 37-40, 3805–3822. MR**2340006**, DOI 10.1016/j.cma.2006.10.043 - Zhengfu Xu,
*Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem*, Math. Comp.**83**(2014), no. 289, 2213–2238. MR**3223330**, DOI 10.1090/S0025-5718-2013-02788-3 - 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 - Qiang Zhang and Chi-Wang Shu,
*Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data*, Numer. Math.**126**(2014), no. 4, 703–740. MR**3175182**, DOI 10.1007/s00211-013-0573-1 - 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 - Xiangxiong Zhang, Yinhua Xia, and Chi-Wang Shu,
*Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes*, J. Sci. Comput.**50**(2012), no. 1, 29–62. MR**2886318**, DOI 10.1007/s10915-011-9472-8 - L. Zhou, Y. Xia, and C-W. Shu,
*Stability analysis and error estimates of arbitrary Lagrangian-Eulerian discontinuous Galerkin method coupled with Runge-Kutta time-marching for linear conservation laws*, ESAIM: Mathematical Modelling and Numerical Analysis, to appear.

## Additional Information

**Pei Fu**- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
- Email: sxfp2013@mail.ustc.edu.cn
**Gero Schnücke**- Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Köln, Germany
- Email: gschnuec@math.uni-koeln.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): April 10, 2018
- Received by editor(s) in revised form: September 29, 2018
- Published electronically: March 5, 2019
- Additional Notes: The third author’s research was supported by NSFC grants No. 11871449 and No. 11471306, and a grant from the Science & Technology on Reliability & Environmental Engineering Laboratory (No. 6142A0502020817).

The third author is the corresponding author. - © Copyright 2019 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 2221-2255 - MSC (2010): Primary 35L65, 65M60
- DOI: https://doi.org/10.1090/mcom/3417
- MathSciNet review: 3957892