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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.

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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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Authors and Affiliations

  1. Institute for Aerodynamics and Gasdynamics, University of Stuttgart, Pfaffenwaldring 21, 70550, Stuttgart, Germany

    Gregor J. Gassner

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Correspondence to Gregor J. Gassner.

Appendices

Appendix A: Dispersion Relation for the P N P M Schemes

In this section, the dispersion relation plots for all combinations of N and M up to 4 are shown. Furthermore, the accuracy results of all combinations for given points per wavelength and the resulting convergence error behavior are listed in Tables 12–19 (Figs. 5, 6, 7, 8, 9).

Fig. 5
figure 5

Plot of the dispersion relation for M=1 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 6
figure 6

Plot of the dispersion relation for M=2 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 7
figure 7

Plot of the dispersion relation for M=3 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 8
figure 8

Plot of the dispersion relation for M=4 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 9
figure 9

Plot of the dissipation relation for M=1 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Table 12 Dispersion error \(\delta:=|{\operatorname{Re}}(\varOmega (K)) - K|\) for the P N P M schemes with M=1 and different N≤M. The most accurate result in each row is marked in bold
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Table 13 Order of convergence of the dispersion error for the P N P M schemes with M=1 and different N≤M
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Table 14 Dispersion error \(\delta:=|{\operatorname{Re}}(\varOmega (K)) - K|\) for the P N P M schemes with M=2 and different N≤M. The most accurate result in each row is marked in bold
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Table 15 Order of convergence of the dispersion error for the P N P M schemes with M=2 and different N≤M
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Table 16 Dispersion error \(\delta:=|{\operatorname{Re}}(\varOmega (K)) - K|\) for the P N P M schemes with M=3 and different N≤M. The most accurate result in each row is marked in bold
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Table 17 Order of convergence of the dispersion error for the P N P M schemes with M=3 and different N≤M
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Table 18 Dispersion error \(\delta:=|{\operatorname{Re}}(\varOmega (K)) - K|\) for the P N P M schemes with M=4 and different N≤M. The most accurate result in each row is marked in bold
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Table 19 Order of convergence of the dispersion error for the P N P M schemes with M=4 and different N≤M
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Appendix B: Dissipation Relation for the P N P M Schemes

In this section, the dissipation relation plots for all combinations of N and M up to 4 are shown. Furthermore, the accuracy results of all combinations for given points per wavelength and the resulting convergence error behavior are listed in Tables 20–27 (Figs. 10, 11, 12).

Fig. 10
figure 10

Plot of the dissipation relation for M=2 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 11
figure 11

Plot of the dissipation relation for M=3 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Fig. 12
figure 12

Plot of the dissipation relation for M=4 for different polynomial degrees N with N≤M. Quantities with superscript ∗ are normalized with the corresponding value N+1

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Table 20 Dissipation error \(\delta:=|{\operatorname{Im}}(\varOmega (K))|\) for the P N P M schemes with M=1 and different N≤M. The most accurate result in each row is marked in bold
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Table 21 Order of convergence of the dissipation error for the P N P M schemes with M=1 and different N≤M
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Table 22 Dissipation error \(\delta:=|{\operatorname{Im}}(\varOmega (K))|\) for the P N P M schemes with M=2 and different N≤M. The most accurate result in each row is marked in bold
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Table 23 Order of convergence of the dissipation error for the P N P M schemes with M=2 and different N≤M
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Table 24 Dissipation error \(\delta:=|{\operatorname{Im}}(\varOmega (K))|\) for the P N P M schemes with M=3 and different N≤M. The most accurate result in each row is marked in bold
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Table 25 Order of convergence of the dissipation error for the P N P M schemes with M=3 and different N≤M
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Table 26 Dissipation error \(\delta:=|{\operatorname{Im}}(\varOmega (K))|\) for the P N P M schemes with M=4 and different N≤M. The most accurate result in each row is marked in bold
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Table 27 Order of convergence of the dissipation error for the P N P M schemes with M=4 and different N≤M
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Gassner, G.J. An Analysis of the Dissipation and Dispersion Errors of the P N P M Schemes. J Sci Comput 54, 21–44 (2013). https://doi.org/10.1007/s10915-012-9600-0

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