Periodic Upwind SBP Operators

• Periodic upwind SBP operators are discrete differential operators on periodic grids that blend upwind bias with built-in dissipation and rigorous SBP properties.
• They are constructed as dual-pairs of forward and backward difference stencils coupled with a positive-definite norm matrix to ensure skew-symmetry and energy stability.
• These operators enable high-order, robust discretizations for hyperbolic conservation laws and kinetic models, effectively handling irregular meshes and cut cells.

A periodic upwind summation-by-parts (SBP) operator is a discrete differential operator defined on periodic grids that generalizes classical central SBP operators by incorporating built-in upwind bias and dissipation, yet maintains rigorous mimetic properties such as discrete integration by parts. These operators are constructed as dual-pairs of forward and backward difference stencils (denoted $D^+$ and $D^-$) in combination with a symmetric positive-definite norm matrix $M$ (or $H$) and are essential in the design of high-order, stable, and asymptotic-preserving numerical schemes for hyperbolic conservation laws and kinetic equations with periodic boundary conditions. The periodic upwind SBP framework, with its explicit structure of skew-symmetry and dissipation, ensures semidiscrete energy-stability, entropy-stability, and robust handling of mesh irregularities such as cut cells (Petri et al., 9 Jan 2026, Duru et al., 2024).

1. Mathematical Definition and Key Properties

Let $u \in \mathbb{R}^N$ denote a discrete solution vector sampled on a uniform periodic grid, and $M \in \mathbb{R}^{N\times N}$ a symmetric positive-definite “norm” matrix that discretizes the $L^2$ inner product. The discrete derivative operator $D$ is said to be a periodic SBP (summation-by-parts) operator if

$MD + D^\top M = 0.$

This skew-symmetry eliminates boundary contributions in periodic settings.

A periodic upwind SBP pair consists of two operators, $D^+$ (forward/upwind) and $D^-$ (backward/downwind), sharing $M$ and satisfying

• Skew-symmetry (“SBP-1”): $MD^+ + (D^-)^\top M = 0$
• Dissipation (“SBP-2”): $M(D^+ - D^-)$ is negative semi-definite

The semi-definite dissipation property introduces built-in numerical dissipation essential for robust upwind discretizations, in contrast to traditional central-SBP operators that lack intrinsic damping mechanisms and localize dissipation in numerical fluxes (Petri et al., 9 Jan 2026, Duru et al., 2024).

2. Construction via Domain-of-Dependence Stabilization

On arbitrary meshes including cut-cell configurations, the domain-of-dependence (DoD) stabilization technique augments a background Discontinuous Galerkin (DG) discretization with local penalty terms to preserve the SBP structure in the presence of geometric irregularities.

For a mesh with a cut cell $E_c$ of size $\alpha\Delta x$, the DoD procedure introduces interface and volume correction terms $J_h^{0,c,\pm}$ and $J_h^{1,c,\pm}$ into the DG bilinear form. For upwind discretizations, these terms alone break the skew-symmetry (SBP-1); a symmetrization is applied:

$$\begin{aligned} \hat a_h^{+,sym}(u,w) &= \hat a_h^z(u,w) - \frac{1}{2}[a_h^{diss}(u,w) + a_h^{diss}(w,u)]\ \hat a_h^{-,sym}(u,w) &= \hat a_h^z(u,w) + \frac{1}{2}[a_h^{diss}(u,w) + a_h^{diss}(w,u)] \end{aligned}$$

where $\hat a_h^z$ is the central (skew-symmetric) form and $a_h^{diss}$ is the dissipation operator. Correspondingly,

$$D^+ = D^z - \frac{1}{2}(D^{diss} + (D^{diss})^\top), \qquad D^- = D^z + \frac{1}{2}(D^{diss} + (D^{diss})^\top)$$

This construction yields periodic upwind SBP operators even on complex meshes (Petri et al., 9 Jan 2026).

3. Dual-Pairing SBP Framework and Periodic Stencil Structure

Within the dual-pairing (DP) SBP finite difference (FD) framework, periodic upwind SBP operators take an explicitly dual form:

$$D^- = H^{-1}(D^+)^\top H, \quad A = H(D^+ - D^-) = A^\top \leq 0,$$

where $H$ is the discrete norm. On a periodic grid, the SBP relation reads

$H D^- + (H D^+)^\top = 0,$

Any resulting boundary term cancels telescopically. A canonical example on a periodic mesh is:

$$D^+ = \Delta x^{-1}[ -\tfrac{1}{2},~\tfrac{3}{2},~ -1,~ \tfrac{1}{2} ] \,\text{(periodic wrap)}, \quad D^- = \Delta x^{-1}[ -\tfrac{1}{2},~1,~ -\tfrac{3}{2},~\tfrac{1}{2} ] \,\text{(periodic wrap)},$$

with $H = \Delta x~I$ (Duru et al., 2024). These satisfy all required SBP properties and are $O(\Delta x^2)$ accurate in the interior.

4. Dissipation, Flux Splitting, and Discrete Stability

The upwind design supplies dissipation everywhere on the mesh, not merely at inter-element boundaries. For nonlinear conservation laws $u_t + f(u)_x = 0$, a flux split

$$f(u) = f^+(u) + f^-(u), \quad f^\pm(u) = \frac{1}{2}[f(u) \pm \gamma g(u)]$$

is employed, where $g(u)$ is typically the entropy variable. The discrete derivative then takes the form

$\partial_x f(u) \approx D^+ f^-(u) + D^- f^+(u).$

This can be rewritten as

$$D^+ f^- + D^- f^+ = D f - \frac{1}{2}(D^+ - D^-)\gamma g, \;\; D = \frac{1}{2}(D^+ + D^-)$$

where the second term induces shock-capturing dissipation of order $O(\Delta x^{2p-1})$. For periodic domains, the SBP identity implies that all boundary terms cancel, ensuring global conservation and no spurious boundary artifacts (Duru et al., 2024).

For kinetic models (e.g., the telegraph equation), the block-system discretized with periodic upwind SBP operators satisfies

$\frac{d}{dt}\|[\rho;g]\|^2_{\underline M} \leq 0,$

so long as the combined block operator is skew-symmetric and the DoD dissipation satisfies $M(D^+ - D^-) \leq 0$ (Petri et al., 9 Jan 2026).

5. Semidiscrete and Fully Discrete Energy and Entropy Stability

For semidiscrete formulations, energy-stability follows from the SBP properties. For discretizations of the telegraph equation with IMEX time-integration and periodic upwind SBP operators,

• The weighted energy is non-increasing in time, with stability derived directly from the skew-symmetry and negative semidefinite dissipation.
• For conservation laws admitting a convex entropy, multiplying the semi-discrete scheme by the entropy variable yields

$$\frac{d}{dt}\sum_j h_j e(u_j) + g^\top H(D^+ f^- + D^- f^+) = 0,$$

and the right-hand term is non-positive, thus the discrete entropy is non-increasing.

For fully discrete schemes using IMEX-RK with asymptotic-preserving splitting, as in the telegraph equation with $\varepsilon \to 0$, the solution automatically collapses to the consistent limit PDE discretization, establishing asymptotic-preservation (Petri et al., 9 Jan 2026).

6. Numerical Performance and Robustness

Numerical experiments demonstrate the robustness and accuracy of periodic upwind SBP operators in multiple regimes:

• Convergence studies reveal $(p+1)$-th order accuracy for alternating-upwind fluxes and predictable stability even for arbitrarily small cut cells ($\alpha \to 10^{-7})$.
• For very stiff kinetic regimes (small scaling parameter $\varepsilon$), schemes remain stable and exhibit $O(\epsilon)$ regularization error in the telegraph-to-heat equation limit (Petri et al., 9 Jan 2026).
• The implicit time discretization of the heat equation on cut-cell meshes is stable only with DoD stabilization, as the implicit solve’s condition number is dramatically improved upon enforcing the SBP properties.

A summary of salient attributes is provided below:

Property	Description	Reference
Built-in dissipation	$M(D^+-D^-)$ negative semi-definite	(Petri et al., 9 Jan 2026, Duru et al., 2024)
Fully periodic	SBP properties enforced for periodic index	(Duru et al., 2024)
Energy/entropy stable	Stability proofs for conservative/kinetic laws	(Petri et al., 9 Jan 2026, Duru et al., 2024)
High-order accuracy	Order $p+1$ for degree $p$ in smooth regimes	(Petri et al., 9 Jan 2026)
Robust to cut cells	Remains accurate/stable as $\alpha\to0$	(Petri et al., 9 Jan 2026)

7. Applications and Generalizations

Periodic upwind SBP operators form the foundation for stable and accurate numerical schemes in a variety of contexts:

• Discontinuous Galerkin (DG) discretizations of kinetic equations—such as the telegraph and linear Boltzmann equations—in periodic domains, especially with challenging geometric configurations like cut cells (Petri et al., 9 Jan 2026).
• High-order, provably entropy-stable finite difference methods for nonlinear hyperbolic systems, including the Burgers, shallow water, and compressible Euler equations, without needing additional nonlinear limiters or post-processing (Duru et al., 2024).
• Extension to multidimensional periodic domains via Kronecker products of 1D operators.

A plausible implication is that this framework enables uniform accuracy, unconditional energy stability, and robust time integration on complex (possibly highly anisotropic) periodic meshes, with potential advantages on next-generation hardware architectures where high-order accuracy and local adaptivity are crucial.

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References

• (Petri et al., 9 Jan 2026) "Domain-of-dependence-stabilized cut-cell discretizations of linear kinetic models with summation-by-parts properties"
• (Duru et al., 2024) "A dual-pairing summation-by-parts finite difference framework for nonlinear conservation laws"