Communications in Applied Mathematics and Computational Science

Discontinuous Galerkin method with the spectral deferred correction time-integration scheme and a modified moment limiter for adaptive grids

Leandro Gryngarten, Andrew Smith, and Suresh Menon

Full-text: Open access

Abstract

The discontinuous Galerkin (DG) method is combined with the spectral deferred correction (SDC) time integration approach to solve the fluid dynamic equations. The moment limiter is generalized for nonuniform grids with hanging nodes that result from adaptive mesh refinement. The effect of characteristic, primitive, or conservative decomposition in the limiting stage is studied. In general, primitive variable decomposition is a better option, especially in two and three dimensions. The accuracy-preserving total variation diminishing (AP-TVD) marker for troubled-cell detection, which uses an averaged-derivative basis, is modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique. Further, a new error estimator is proposed to determine where to refine or coarsen the grid. This estimator is compared against other estimator used in the literature and shows an improved performance. Canonical tests in one, two, and three dimensions are conducted to show the accuracy of the solver.

Article information

Source
Commun. Appl. Math. Comput. Sci., Volume 7, Number 2 (2012), 133-174.

Dates
Received: 15 December 2010
Revised: 8 April 2012
Accepted: 16 April 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.camcos/1513732055

Digital Object Identifier
doi:10.2140/camcos.2012.7.133

Mathematical Reviews number (MathSciNet)
MR3005736

Zentralblatt MATH identifier
06130512

Subjects
Primary: 35L65: Conservation laws 35L67: Shocks and singularities [See also 58Kxx, 76L05] 65L06: Multistep, Runge-Kutta and extrapolation methods 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

Keywords
discontinuous Galerkin moment limiter high-order accuracy adaptive mesh troubled-cell detector spectral deferred correction

Citation

Gryngarten, Leandro; Smith, Andrew; Menon, Suresh. Discontinuous Galerkin method with the spectral deferred correction time-integration scheme and a modified moment limiter for adaptive grids. Commun. Appl. Math. Comput. Sci. 7 (2012), no. 2, 133--174. doi:10.2140/camcos.2012.7.133. https://projecteuclid.org/euclid.camcos/1513732055


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