DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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Periodic diagonal-norm SBP compressible viscous scheme

Definition

Let $I = {1, \dots, M}$ be a nonempty finite periodic node set. Let $H \in R^{M \times M}$ be diagonal with strictly positive entries, and for each direction $j = 1, \dots, d$ let $D_{j} \in R^{M \times M}$ satisfy the periodic diagonal-norm SBP identities

$H D_{j} + D_{j}^{T} H = 0, D_{j} 1 = 0.$

For a nodal array $w = (w_{i})_{i \in I}$ , $D_{j} w$ means componentwise matrix multiplication, $(D_{j} w)_{i} = \sum_{ℓ \in I} (D_{j})_{i ℓ} w_{ℓ}$ . Given admissible nodal states $U_{i} \in G$ , write $v_{i} = v (U_{i})$ . Define the semidiscrete system

${\dot{U}}_{i} + 2 \sum_{j = 1}^{d} \sum_{ℓ \in I} (D_{j})_{i ℓ} F_{j}^{e c} (U_{i}, U_{ℓ}) = \sum_{j, k = 1}^{d} {[D_{j} ((K_{j k} (U_{i}) (D_{k} v)_{i})_{i \in I})]}_{i}, i \in I,$

where $F_{j}^{e c}$ is the two-point flux in Explicit entropy-conservative two-point Euler flux and $K_{j k}$ is the viscous block matrix in Positive-semidefinite entropy-variable factorization of viscous flux. This definition specifies a periodic semidiscrete operator on $G^{M}$ ; it does not include a time discretization or a positivity limiter.

Depends on

Used by

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Sources

Eitan Tadmor, Entropy Stable Schemes, Chapter 18, Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues, 2016, complete chapter (standard reference, not scraped)
Pablo Fernandez, Ngoc-Cuong Nguyen, and Jaime Peraire, Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations, complete author preprint (standard reference, not scraped)