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Summation-by-parts operators for correction procedure via reconstruction

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arxiv 1511.02052 v2 pith:IGIQ5EUE submitted 2015-11-06 math.NA cs.NA

Hendrik Ranocha , Philipp Öffner , Thomas Sonar This is my paper
classification math.NAcs.NA
keywords methodsconservationcorrectionframeworkreconstructionspectralstabilitydifference
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The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.

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Cited by 5 Pith papers

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  1. Computing Radially-Symmetric Solutions of the Ultra-Relativistic Euler Equations with Entropy-Stable Discontinuous Galerkin Methods

    math.NA 2025-08 accept novelty 7.0 of 10

    The authors derive an entropy-conservative two-point flux for the ultra-relativistic Euler equations, prove its consistency, and validate an entropy-stable DG scheme against 1D radial reference solutions in 2D and 3D.

  2. Convergence of entropy-conservative summation-by-parts discretizations to smooth solutions of hyperbolic conservation laws

    math.NA 2026-07 accept novelty 6.0 of 10

    Entropy-conservative diagonal-norm SBP flux-differencing schemes converge at order p to smooth solutions of general entropy-symmetrizable hyperbolic systems under periodic boundaries.

  3. GPU-Accelerated Energy-Conserving Methods for the Two-Dimensional Hyperbolized Serre-Green-Naghdi Equations

    math.NA 2026-01 accept novelty 6.0 of 10

    2D energy-conserving finite-difference SBP/split-form schemes for the hyperbolic Serre-Green-Naghdi equations are derived, proved, and GPU-accelerated in Julia.

  4. Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

    math.NA 2025-07 accept novelty 6.0 of 10

    The semi-discrete Active Flux method for 1D linear advection with periodic boundaries is shown to be energy stable via newly constructed, including degenerate, summation-by-parts operators.

  5. Justification and structure- and asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation

    math.NA 2026-06 unverdicted novelty 4.0 of 10

    Develops energy-stable asymptotic-preserving discretizations of a hyperbolized Cahn-Hilliard equation via SBP operators and IMEX Runge-Kutta methods guided by relative-energy error estimates.

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