A space-time discontinuous Galerkin method for linear hyperbolic PDE's with high frequencies

Year 2020, Volume: 69 Issue: 1, 213 - 231, 30.06.2020

Şuayip Toprakseven

https://doi.org/10.31801/cfsuasmas.544522

Abstract

The main purpose of this paper is to describe a space-time discontinuous Galerin (DG) method based on an extended space-time approximation
space for the linear first order hyperbolic equation that contains a high frequency component. We extend the space-time DG spaces of tensor-product of
polynomials by adding trigonometric functions in space and time that capture
the oscillatory behavior of the solution. We construct the method by combining the basic framework of the space-time DG method with the extended finite
element method. The basic principle of the method is integrating the features
of the partial differential equation with the standard space-time spaces in the
approximation. We present error analysis of the space-time DG method for
the linear first order hyperbolic problems. We show that the new space-time
DG approximation has an improvement in the convergence compared to the
space-time DG schemes with tensor-product polynomials. Numerical examples verify the theoretical findings and demonstrate the effects of the proposed
method.

Keywords

Discontinuous Galerkin finite element methods, space-time discontinuous Galerkin methods, hyperbolic problems, high frequency solutions

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