Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws (Communications in Computational Physics, 2018)
In this paper, we consider high order multi-domain penalty spectral Galerkinmethods for the approximation of hyperbolic conservation laws. This formulation hasa penalty parameter which can vary in space and time, allowing for flexibility in thepenalty formulation. This flexibility is particularly advantageous for problems withan inhomogeneous mesh. We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value ofthe penalty parameter. The penalty parameter has an effect on both the accuracy andstability of the method. We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods. The coefficienttruncation method is proposed to prevent the rapid error growth due to round-offerrors when high order polynomials are used. Finally, we show that an inconsistentevaluation of the integrals in the penalty method may lead to growth of errors. Numerical examples for linear and nonlinear problems are presented.