# An Analysis of the Dissipation and Dispersion Errors of the *P* _{ N } *P* _{ M } Schemes

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## Abstract

We examine the dispersion and dissipation properties of the *P* _{ N } *P* _{ M } schemes for linear wave propagation problems. *P* _{ N } *P* _{ M } scheme are based on *P* _{ N } discontinuous Galerkin base approximations augmented with a cell centered polynomial least-squares reconstruction from degree *N* up to the design polynomial degree *M*. This methodology can be seen as a generalized discretization framework, as cell centered high order finite volume schemes (*N*=0) and discontinuous Galerkin schemes (*N*=*M*) are included as special cases.

We show that with respect to the dispersion error, the pure discontinuous Galerkin variant gives typically the best accuracy for a defined number of points per wavelength. Regarding the dissipation behavior, combinations of *N* and *M* exist that result in slightly lower errors for a given resolution. An investigation of the influence of the stencil size on the accuracy of the scheme shows that the errors are smaller the smaller the stencil size for the reconstruction.

## Keywords

Dispersion Dissipation Reconstructed discontinuous Galerkin*P*

_{N}

*P*

_{M}schemes Wave propagation

## Notes

### Acknowledgements

The author thanks the reviewers for their valuable comments. This project is kindly supported by the Deutsche Forschungsgemeinschaft (DFG) within SPP 1276: MetStroem and the research project IDIHOM within the European Research Framework Programme.

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