arXiv subjects

Mathias Dufresne-Piché

Publications and source records attributed to Mathias Dufresne-Piché.

3 recordsLinked to original sources

arXiv2026

On the Role of Split Formulations on Aliasing Errors and Entropy Stability of Discontinuous Galerkin Schemes

In this work, we formally investigate the dealiasing properties of split form discontinuous Galerkin (DG) discretizations for the one-dimensional Burgers problem. By generalizing the proof of Blaisdell et al. for spectral discretizations, we show that split form DG schemes achieve dealiasing in the weakly underresolved range through integration error cancellation on the conservative and non-conservative forms. As a corollary, we identify quadrature- and order-dependent pairs of splitting coefficients that eliminate the dominant component of the aliasing error. While these ``optimized'' splitting coefficients minimize integration errors on the numerical scheme, we show that the skew-symmetric aliasing pattern resulting from the entropy stable split is required to maintain long-term stability of the numerical solution. The proposed framework provides an alternative proof for the entropy stability of the DG discretization of the (1/3, 2/3) split formulation introduced by Gassner which clarifies the connection between DG aliasing errors and entropy stability. Finally, we also show that the entropy stable split is associated with lower aliasing errors in the weakly underresolved range compared to the conservative form.

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arXiv2025

On the Superconvergence of ESFR Schemes

The energy stable flux reconstruction (ESFR) method provides an efficient and flexible framework to devise high-order linearly stable numerical schemes which can achieve high levels of accuracy on unstructured grids. While superconvergent properties of ESFR schemes have been observed in numerical experiments, no formal proof of this behavior has been reported in the literature. In this work, we attempt to address this by providing a simple derivation for the superconvergence of the dispersion-dissipation error of ESFR schemes for the linear advection problem when using an upwind numerical flux. We show that the superconvergence of ESFR schemes essentially relies on the capacity of the latter to generate superconvergent rational approximants of the exponential function, which is reminiscent of well-known theoretical results for superconvergence of discontinuous Galerkin (DG) methods. We also demonstrate that the drops in order of accuracy which are observed in numerical experiments as the ESFR scalar $c$ is increased are caused by both a modification of the structure of these rational approximants and a change in the multiplicity of the physical eigenvalue of the schemes as $c \to \infty$. Finally, our theoretical results are successfully validated against numerical experiments.

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arXiv2025

Linearly Stable Generalizations of ESFR Schemes

The energy stable flux reconstruction (ESFR) method encompasses an infinite family of high-order, linearly stable schemes and thus provides a flex- ible and efficient framework for achieving high levels of accuracy on unstruc- tured grids. One remarkable property of ESFR schemes is their ability to be expressed equivalently as linearly filtered discontinuous Galerkin (FDG) schemes. In this study, we introduce Sobolev Stable discontinuous Galerkin (SSDG) schemes, a new conservative and linearly stable generalization of ESFR schemes via the FDG framework. Additionally, we review existing generalizations of the ESFR method and consider their relationship with the FDG framework. The linear properties of SSDG schemes are studied via Von Neumann analysis and compared to those of the existing extended ESFR (EESFR) method. It is found that while SSDG and EESFR schemes exhibit fundamentally different dispersive and dissipative behaviors, they can achieve a similar increase in CFL limit and exhibit a similar spectral order of accuracy. Moreover, it is seen that the range of scheme parameters over which SSDG schemes can be used to increase the explicit time-stepping limit is much larger than for EESFR schemes. Finally, it is observed that the order of accuracy of EESFR schemes under h-refinement is generally p + 1 while that of SSDG schemes is p.

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