---
title: Periodic Upwind SBP Operators
url: https://www.emergentmind.com/topics/periodic-upwind-sbp-operators
type: topic
---

# Periodic Upwind SBP Operators

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 [2601.05817][2411.06629].

## 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 [2601.05817][2411.06629].

## 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 [2601.05817].

## 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 $H = \Delta x~I$ [2411.06629]. 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 [2411.06629].

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 $M(D^+ - D^-) \leq 0$ [2601.05817].

## 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 [2601.05817].

## 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 [2601.05817].
- 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           | [2601.05817][2411.06629] |
| Fully periodic       | SBP properties enforced for periodic index    | [2411.06629]         |
| Energy/entropy stable| Stability proofs for conservative/kinetic laws| [2601.05817][2411.06629] |
| High-order accuracy  | Order $p+1$ for degree $p$ in smooth regimes  | [2601.05817]         |
| Robust to cut cells  | Remains accurate/stable as $\alpha\to0$       | [2601.05817]         |

## 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 [2601.05817].
- 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 [2411.06629].
- 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.

---

**References**

- [2601.05817] "Domain-of-dependence-stabilized cut-cell discretizations of linear kinetic models with summation-by-parts properties"
- [2411.06629] "A dual-pairing summation-by-parts finite difference framework for nonlinear conservation laws"

Source: https://www.emergentmind.com/topics/periodic-upwind-sbp-operators