Tensor-Product Split-Simplex Summation-By-Parts Operators

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   Résumé scientifique

   Summation-by-parts (SBP) operators are a cornerstone of provably stable high-order methods for hyperbolic conservation laws. For nonlinear problems, the flux-differencing framework of Fisher and Carpenter combines such SBP operators with entropy-conservative two-point fluxes to obtain semi-discretizations that are conservative in the sense of Lax--Wendroff and discretely entropy-conservative or stable. So far, this framework has been developed exclusively for operators that are exact for polynomials. Function-space SBP (FSBP) operators lift this restriction by being exact on a general finite-dimensional function space that can be tailored to the problem at hand. In this work, we extend flux …

2. Proche Pas pertinent

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   Zelalem Arega Worku, D. C. Del Rey Fernández · 2026 · cité 1 fois · Pas encore résumé

   Résumé scientifique

   Residual correction provides an efficient alternative to two-point volume flux differencing for enforcing entropy stability in high-order discretizations of hyperbolic partial differential equations. We develop an element-local residual-correction summation-by-parts (SBP) method that combines continuous-SBP (C-SBP) and discontinuous-SBP (D-SBP) coupling on conforming meshes. The construction retains continuous assembly without simultaneous approximation terms (SATs) wherever suitable, restricts SATs to selected discontinuous facets, and permits the coupling to change in time as a foundation for adaptive discretizations. The scalar entropy defect and a graph-based distribution vector acting o…

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   Résumé scientifique

   High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from $\mathcal{O}(N^d)$ to $\mathcal{O}(dNr)$. TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral trunca…

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   Résumé scientifique

   A common approach to representing disk-like and sphere-like geometries in isogeometric analysis is to use collapsed-edge singularities in the underlying parameterization. The resulting tensor-product B-spline spaces on such singular parameterizations lack the required smoothness to be used in isogeometric analysis. Polar splines (Toshniwal et al., CMAME 2017) rectify this issue by extracting a smoother subspace, and have been successfully used to discretize high-order partial differential equations, as well as to perform structure-preserving discretizations of the de Rham complex.The latter is particularly relevant for mixed formulations that appear in applications such as electromagnetism a…

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   Résumé scientifique

   We present an interpretation of the fast matrix multiplication algorithms of Coppersmith and Winograd as aggregations of tensor contractions. This point of view provides a direct connection to older constructions by Pan and to more recent work on bilinear algorithms for symmetric tensor contraction. Further, we provide a simple explicit approach for exact aggregation of tensor contractions, each one yielding an independent matrix-matrix product. These constructions and border rank adaptations thereof do not come close to improving the best known bounds on $\omega$ (the matrix multiplication exponent), but provide explicit moderate-size fast matrix multiplication algorithms that achieve a bet…

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   Résumé scientifique

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   Résumé scientifique

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   Résumé scientifique

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Les articles qui le citent

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   A collocation scheme that is equivalent to discontinuous Galerkin discretizations

   Jason E. Hicken · 2026 · arXiv.org · preprint, non relu par les pairs · cité 1 fois · Pas encore résumé

   Résumé scientifique

   A spectral collocation operator with the summation-by-parts property was introduced by Chan to develop entropy-stable discontinuous Galerkin (DG) semi-discretizations (https://doi.org/10.1016/j.jcp.2018.02.033). The present work shows that semi-discretizations based on this collocation operator produce solutions that are equivalent to solutions of a DG semi-discretization using the same underlying quadrature. The equivalence holds regardless of the number of degrees of freedom in the collocation scheme and when the quadrature is not strictly positive. Extraneous degrees of freedom in the collocation scheme are associated with the nullspace of the operator and remain zero throughout an unstea…

2. Apparenté Pas pertinent

   Finite-difference compatible entropy-conserving schemes for the compressible Euler equations

   C. De Michele, A. Edoh, G. Coppola · 2024 · Journal of Computational Physics · cité 10 fois · Pas encore résumé

   Résumé scientifique

   This paper introduces a family of entropy-conserving finite-difference discretizations for the compressible flow equations. In addition to conserving the primary quantities of mass, momentum, and total energy, the methods also preserve kinetic energy and pressure equilibrium. The schemes are based on finite-difference (FD) representations of the logarithmic mean, establishing and leveraging a broader link between linear and nonlinear two-point averages and FD forms. The schemes are locally conservative due to the summation-by-parts property and therefore admit a local flux form, making them applicable also in finite-volume and finite-element settings. The effectiveness of these schemes is va…