Residual based error estimates for the space-time discontinuous Galerkin method applied to nonlinear hyperbolic equations

Main Article Content

Vít Dolejší Filip Roskovec

Abstract

We present an adaptive numerical method for solving nonlinear hyperbolic equations.  The method uses the space-time discontinuous Galerkin discretization, exploiting its high polynomial approximation degrees with respect to both space and time coordinates.  We derive an residual-based a posteriori error estimator and propose an efficient strategy how to identify the parts of the computational error caused by the space and time discretization, respectively, as well as the errors arising from the linearization of the resultant algebraic system of equations. Further, an algorithm keeping all these three components of the computational error balanced is presented. The computational performance of the proposed method is demonstrated by numerical experiments.

Article Details

How to Cite
DOLEJŠÍ, Vít; ROSKOVEC, Filip. Residual based error estimates for the space-time discontinuous Galerkin method applied to nonlinear hyperbolic equations. Proceedings of the Conference Algoritmy, [S.l.], p. 113-124, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/400>. Date accessed: 22 sep. 2017.
Section
Articles

References

[1] L. El Alaoui, A. Ern, and M. Vohral ́ık. Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Methods Appl. Mech. Engrg, 200:2782–2795, 2011.

[2] E. Burman and A. Ern. Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian. Comptes Rendus Mathematique, 346(17-18):1013–1016, 2008.

[3] V. Dolejší. Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci., 1(3):165–178, 1998.

[4] V. Dolejší. Semi-implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows. Commun. Comput. Phys., 4(2):231–274, 2008.

[5] V. Dolejší. A design of residual error estimates for a high order BDF-DGFE method applied to compressible flows. Int. J. Numer. Meth. Fluids, 73(6):523–559, 2013.

[6] V. Dolejší. hp-DGFEM for nonlinear convection-diffusion problems. Math. Comput. Simul., 87:87–118, 2013.

[7] V. Dolejší and M. Feistauer. Semi-Implicit Discontinuous Galerkin Finite Element Method for the Numerical Solution of Inviscid Compressible Flow. J. Comput. Phys., 198(2):727–746, 2004.

[8] V. Dolejší and M. Feistauer. Discontinuous Galerkin Method, volume 48. Springer International Publishing, 2015.

[9] V. Dolejší, M. Holík, and J. Hozman. Efficient solution strategy for the semi-implicit discontinuous Galerkin discretization of the Navier-Stokes equations. J. Comput. Phys., 230:4176– 4200, 2011.

[10] V. Dolejší, F. Roskovec, and M. Vlasák. Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows. Computers & Fluids, 117:304–324, 2015.

[11] M. Feistauer, V. Kučera, K. Najzar, and J. Prokopová. Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Math., 117:251–288, 2011.

[12] E. Hairer and G. Wanner. Solving ordinary differential equations II, Stiff and differentialalgebraic problems. Springer Verlag, 2002.

[13] A. Kufner, O. John, and S. Fučík. Function Spaces. Academia, Prague, 1977.

[14] J. Nečas. Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967.

[15] Deuflhard P. Newton Methods for Nonlinear Problems. Springer Series in Computational Mathematics, Vol. 35, 2004.

[16] C.W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In A. Quarteroni et al, editor, Advanced numerical approximation of nonlinear hyperbolic equations, Lect. Notes Math. 1697, pages 325–432. Berlin: Springer, 1998.

[17] R. Verfurth. A Posteriori Error Estimates for Nonlinear Problems: Lr (0, T ; Lρ (Ω))-Error Estimates for Finite Element Discretizations of Parabolic Equations. Math. Comput., 67(224):1335–1360, 1998.

[18] R. Verfurth. A Posteriori Error Estimates for Nonlinear Problems: Lr (0, T ; W 1,ρ (Ω))-Error Estimates for Finite Element Discretizations of Parabolic Equations. Numer. Meth. Part. Diff. Eqs, 14:487–518, 1998.