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Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties

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Abstract

The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators. Furthermore, LPS is able to take advantage of the additional nodes required by degree 2p diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.

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Notes

  1. Assume the Legendre polynomials are normalized to have unit \(L^2\) norm.

  2. Again, the bound requires the assumption that the density and pressure remain positive.

  3. In this work, \(\textsf {H}_{\kappa }\) is 2p exact, but, as was shown in [29], one only needs \(2p-1\) exactness for the operators to exist.

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Acknowledgements

My sincerest thanks to my students—Anthony Ashley, Tucker Babcock, Garo Bedonian, Luiz Cagliari, Jared Crean, Sharanjeet Kaur, Kinshuk Panda, Ge Yan, and Jianfeng Yan—for their feedback on an early draft of this paper. I am also grateful to the anonymous reviewers for their careful reading of the manuscript and helpful comments. All the results in this paper were obtained using software written in Julia [4]. The plots were generated using Matplotlib [35], with help from the Numpy [42, 50] and Scipy [38] libraries.

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    Jason E. Hicken

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Appendix: Eigenvalues and eigenvectors of the LPS operator

Appendix: Eigenvalues and eigenvectors of the LPS operator

Theorem 4

Consider an element in dimension d, whose \(n_{\kappa }\) nodes and norm \(\textsf {H}_{\kappa }\) define a degree 2p exact cubature rule. Let \(\textsf {L}\) denote the orthogonal polynomials of total degree p on the element domain evaluated at the nodes of the element, and define the LPS dissipation operator \(\textsf {M}_{\kappa }^{\textsf {P}}= \left( \textsf {P}_{\kappa }\right) ^T \textsf {H}_{\kappa }\textsf {P}_{\kappa }\), with unit scaling matrix, \(\textsf {A}_{\kappa } = \textsf {I}\). Then the eigenvalues of \(\textsf {H}_{\kappa }^{-1} \textsf {M}_{\kappa }^{\textsf {P}}\) are \(\lambda _0 = 0\), with multiplicity \({p + d} \atopwithdelims (){d}\), and \(\lambda _1 =1\), with multiplicity \(n_{\kappa }- {{p + d} \atopwithdelims (){d}}\). The eigenvectors corresponding to \(\lambda _0\) are \(\textsf {L}\) and the eigenvectors corresponding to \(\lambda _1\) are in the nullspace of \(\textsf {L}^T \textsf {H}_{\kappa }\).

Proof

The first step in the proof is to find the eigenvalues of the projection operator, \(\textsf {P}_{\kappa }= \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\). It is easy to show that the columns of \(\textsf {L}\) are eigenvectors of \(\textsf {P}_{\kappa }\) corresponding to eigenvalues \(\lambda _0=0\):

$$\begin{aligned} \textsf {P}_{\kappa }\textsf {L} = \textsf {L} - \textsf {L} \left( \textsf {L}^T \textsf {H}_{\kappa }\textsf {L} \right) = \textsf {L} - \textsf {L} = \textsf {0}, \end{aligned}$$

where we used the orthogonality of the \(\textsf {L}\) and the exactness of the cubature for polynomials of degree 2p or less. Now, let \(\textsf {V}\) denote an orthonormal basis for the nullspace of \(\textsf {L}^T \textsf {H}_{\kappa }\). Since there are \({p + d} \atopwithdelims (){d}\) linearly independent vectors in \(\textsf {L}\), and \(\textsf {H}_{\kappa }\) is full rank, the nullspace has dimension \(n_{\kappa }- {{p + d} \atopwithdelims (){d}}\). Furthermore, the columns of \(\textsf {V}\) are also eigenvectors of \(\textsf {P}_{\kappa }\) with associated eigenvalues \(\lambda _1=1\):

$$\begin{aligned} \textsf {P}_{\kappa }\textsf {V} = \textsf {V} - \textsf {L} \left( \textsf {L}^T \textsf {H}_{\kappa }\textsf {V} \right) = \textsf {V}, \end{aligned}$$

since \(\textsf {V}\) is in the nullspace of \(\textsf {L}^T \textsf {H}_{\kappa }\).

The second, and final, step in the proof is to show that \(\textsf {H}_{\kappa }^{-1} \textsf {M}_{\kappa }^{\textsf {P}}= \textsf {P}_{\kappa }^2\), from which it follows that the desired eigenvalues are the square of those of \(\textsf {P}_{\kappa }\). To this end,

$$\begin{aligned} \textsf {H}_{\kappa }^{-1} \textsf {M}_{\kappa }^{\textsf {P}}= \textsf {H}_{\kappa }^{-1} \left( \textsf {P}_{\kappa }\right) ^T \textsf {H}_{\kappa }\textsf {P}_{\kappa }= & {} \textsf {H}_{\kappa }^{-1} \left( \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\right) ^T \textsf {H}_{\kappa }\left( \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\right) \\= & {} \textsf {H}_{\kappa }^{-1} \left( \textsf {I} - \textsf {H}_{\kappa }\textsf {L} \textsf {L}^T \right) \textsf {H}_{\kappa }\left( \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\right) \\= & {} \left( \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\right) \left( \textsf {I} - \textsf {L} \textsf {L}^T \textsf {H}_{\kappa }\right) = \textsf {P}_{\kappa }\textsf {P}_{\kappa }, \end{aligned}$$

where we used the symmetry of \(\textsf {H}_{\kappa }\) in the second last line above. Thus, based on this result and the eigenvalues of \(\textsf {P}_{\kappa }\), we must have \(\left( \textsf {H}_{\kappa }^{-1} \textsf {M}_{\kappa }^{\textsf {P}}\right) \textsf {L} = \textsf {0}\) and \(\left( \textsf {H}_{\kappa }^{-1} \textsf {M}_{\kappa }^{\textsf {P}}\right) \textsf {V}\) = \(\textsf {V}\), which completes the proof. \(\square \)

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Hicken, J.E. Entropy-Stable, High-Order Summation-by-Parts Discretizations Without Interface Penalties. J Sci Comput 82, 50 (2020). https://doi.org/10.1007/s10915-020-01154-8

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