Commun. Comput. Phys., 5 (2009), pp. 600-619.


Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws

Sigal Gottlieb 1, Jae-Hun Jung 2*

1 Department of Mathematics, University of Massachusetts at Dartmouth, North Dartmouth, MA 02747-2300, USA.
2 Department of Mathematics, University of Massachusetts at Dartmouth, North Dartmouth, MA 02747-2300, USA; and Department of Mathematics, SUNY at Buffalo, Buffalo, NY, 14260-2900, USA.

Received 3 October 2007; Accepted (in revised version) 10 December 2007
Available online 1 August 2008

Abstract

In this paper, we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws. This formulation has a penalty parameter which can vary in space and time, allowing for flexibility in the penalty formulation. This flexibility is particularly advantageous for problems with an inhomogeneous mesh. We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter. The penalty parameter has an effect on both the accuracy and stability of the method. We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods. The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used. Finally, we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors. Numerical examples for linear and nonlinear problems are presented.

AMS subject classifications: 65M70, 65M12, 65M60, 65L07

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Key words: High order polynomial Galerkin methods, penalty boundary conditions, discontinuous Galerkin methods, hyperbolic conservation laws, round-off errors, truncation methods.

*Corresponding author.
Email: sgottlieb@umassd.edu (S. Gottlieb), jjung@umassd.edu (J.-H. Jung)
 

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