Explicit Staggered Scheme Overview

• Explicit staggered scheme is a numerical method that arranges primary variables on interlaced grids to ensure exact algebraic conservation and suppress spurious oscillations.
• It uses time-explicit integrators such as leap-frog and Runge–Kutta to achieve second-order accuracy while adhering to strict CFL stability constraints.
• The approach enhances parallel efficiency and decoupling, making it ideal for multi-physics applications in fluid dynamics, transport, and network simulations.

An explicit staggered scheme is a class of time-explicit numerical methods that employ a staggered arrangement of variables across space and/or time grids to discretize partial differential equations, particularly in computational fluid dynamics, transport, dissipative systems, multi-physics coupling, and related applications. This design enables exact algebraic conservation, enhanced stability, and high-order accuracy while avoiding large coupled linear or nonlinear solves. The architecture contrasts with collocated schemes by placing conjugate variables (e.g., density and flux, or pressure and velocity) at alternating spatial or temporal locations. Such schemes are widely used for compressible/incompressible flows, shallow-water or multiphase models, and network problems, and are subject to explicit CFL stability constraints. Below, the fundamental principles, mathematical construction, algorithmic advantages, and representative applications of explicit staggered schemes are surveyed with precise reference to contemporary research literature.

1. Mathematical Structure and Staggered Grid Arrangement

In explicit staggered schemes, primary variables are distributed in a non-collocated manner over the discretized domain, typically on interlaced (“primal” and “dual”) grids. For example, in one-dimensional pipe flow simulation, density and pressure are set at cell centers (primal nodes) $x_i = i\Delta x$, while mass flux (or velocity) is placed at edge midpoints (dual nodes) $x_{j} = x_{i + 1/2} = (i+1/2)\Delta x$ (Gyrya et al., 2018). Time integration may also use temporal staggering, e.g., storing densities at integer time steps $t^n$ and fluxes at half-steps $t^{n+1/2}$. This mirrors the approach in explicit leap-frog or leap-frog-like schemes for the (2+1)D Dirac equation and fluctuating hydrodynamics (Hammer et al., 2013, Balboa et al., 2011).

This grid arrangement directly supports conserved discretization of continuity, momentum, and auxiliary relations:

• Mass continuity (e.g., in pipeline or shallow water flows):

$$\frac{\rho_i^{n+1} - \rho_i^n}{\Delta t} + \frac{\phi_{j}^{n+1/2} - \phi_{j-1}^{n+1/2}}{\Delta x} = 0$$

• Momentum update (on dual grid):

$$\frac{\phi_{j}^{n+1} - \phi_{j}^{n}}{\Delta t} + \frac{p_{i+1}^{n+1} - p_{i}^{n+1}}{\Delta x} = \cdots$$

This structure guarantees local conservation and leads to discrete updates that require only scalar (local) inversions, not large global solves, as in implicit or monolithic schemes (Gyrya et al., 2018, Seifi et al., 2018, Srivastava et al., 2022). The spatial staggering also naturally suppresses spurious pressure and velocity oscillations (“checkerboarding”) in finite-difference and finite-volume frameworks.

2. Explicit Time Integration and Stability Constraints

Explicit staggered schemes update the solution by evaluating all right-hand sides using previous- or staggered-time-level data, avoiding implicit solves or subiterations. For each primary variable, a forward Euler, leap-frog, or higher-order explicit Runge–Kutta integrator advances the solution in time:

• Second-order leap-frog: pairs of variables are updated at interleaved time levels, yielding formal $O(\Delta t^2)$ accuracy (e.g., leap-frog for Dirac/Weyl, pipeline, and fluctuating hydrodynamics (Hammer et al., 2013, Gyrya et al., 2018, Balboa et al., 2011)).
• Explicit Runge–Kutta: e.g., three-stage SSP RK3 schemes for stochastic hydrodynamics on staggered grids (Balboa et al., 2011).

The explicit nature imposes strict Courant–Friedrichs–Lewy (CFL) constraints for stability:

• Wave CFL: e.g., for nonlinear gas dynamics,

  $$\max_j \left[ c(\rho^{n+1}_j) \right] \cdot \frac{\Delta t}{\Delta x} \leq 1$$

  where $c(\rho)$ is a (possibly composition- and state-dependent) wave speed (Gyrya et al., 2018, Brodskyi et al., 2024).

• Diffusive CFL: $\Delta t \leq C \Delta x^2$ for dominant parabolic terms (Küpper et al., 2015).
• Hybrid (Multiphysics) CFL: the more restrictive bound, coupling both hyperbolic and parabolic terms, applies in AP transport (Küpper et al., 2015).

This conditional stability regime is the central limitation of the explicit staggered scheme, dictating time-step size for the desired spatial resolution and system eigenstructure (Gyrya et al., 2018, Balboa et al., 2011).

3. Algorithmic Advantages, Conservation, and Order

Explicit staggered schemes offer several technical strengths:

• Exact discrete conservation: The fluxes at dual/primal interfaces are constructed so that the sum of inflow and outflow exactly balances the rate of change of conserved quantities, up to machine roundoff (Gyrya et al., 2018, Srivastava et al., 2022). Mass, momentum, and energy (or their numerical analogs) are preserved to machine precision in the absence of truncation error sources (e.g., boundary treatments).
• Second-order accuracy: Centered finite differences, together with temporal staggering, yield $O(\Delta x^2 + \Delta t^2)$ truncation error for smooth solutions (Gyrya et al., 2018, Hammer et al., 2013).
• Local update and high parallel efficiency: No global, coupled system must be solved at each step; only scalar or small local nonlinear equations (e.g., quadratic for friction in pipeline flow) need inversion, and all nodes/cells can be updated concurrently (Gyrya et al., 2018).
• Decoupling and positivity: In dissipative systems and auxiliary variable formulations (e.g., energy-stable schemes), the staggered approach permits linear, fully decoupled updates that are unconditionally energy dissipation preserving and positivity preserving for the system energy-like variables (Liu et al., 14 Mar 2025).
• Shock-compatibility and weak solution convergence: Correction strategies within the staggered framework can restore exact conservation of momentum and total energy, ensuring convergence to weak (possibly discontinuous) solutions of hyperbolic conservation laws under mesh refinement (Abgrall, 2021).

However, shock-resolving capabilities are limited: strong discontinuities may not be robustly captured unless further stabilization (e.g., artificial viscosity, flux limiters) is introduced.

4. Boundary, Junction, and Multi-Physics Coupling

The explicit staggered structure is compatible with complex boundary and networked system treatments:

• Boundary conditions: Dirichlet and Neumann/Robin boundary conditions are enforced directly at staggered locations, often via projection or nonlinear scalar solves that preserve conservation (Gyrya et al., 2018, Hammer et al., 2013, Srivastava et al., 2022).
• Network junctions: Flow and pressure (or density) values at network nodes are enforced by nonlinear scalar compatibility conditions, incorporating effects like compressor pressure jumps via discrete equations derived from mass and momentum continuity and equation of state inversion (Gyrya et al., 2018, Brodskyi et al., 2024).
• Multi-physics and operator splitting: Staggered explicit-implicit (partitioned) schemes for coupled fluid–structure interaction alternate explicit updates for one field (structure, mechanics) with implicit updates for another (fluid, electrostatics), with Robin–Robin interface conditions enabling interaction without iterative subcycling (Guo et al., 15 Oct 2025, Seifi et al., 2018). Extensions to three-step procedures handle internal variables for dissipative processes (e.g., plasticity, damage) (Roubíček et al., 2020).

This dimension allows for efficient simulations in complex domain topologies and multiphysics settings, facilitating real-world applications such as gas pipeline networks with active monitoring and control (Brodskyi et al., 2024).

5. Applications and Case Studies

Explicit staggered schemes have been constructed and validated in multiple contexts:

Application	Scheme Structure	Reference
Gas pipeline networks (single/multiphase)	Second-order explicit FD, leap-frog staggering	(Gyrya et al., 2018, Brodskyi et al., 2024)
Fluctuating hydrodynamics (stochastic PDEs)	Explicit finite-volume, momentum face-staggered	(Balboa et al., 2011, Srivastava et al., 2022)
Dirac/Weyl equation (quantum transport)	Leap-frog in time, spatial-time spinor staggering	(Hammer et al., 2013)
Shallow water equations	Staggered finite-volume, edge velocity, MAC grid	(Herbin et al., 2021)
Compressible Euler (general construction)	Element-wise residual correction, arbitrary degree	(Abgrall, 2021)
Diffusive AP-transport (multiscale limits)	Parity variables, staggered stencil, time splitting	(Küpper et al., 2015)
Coupled multiphysics (FSI, electromechanics)	Explicit–implicit staggered, partitioned time stepping	(Seifi et al., 2018, Guo et al., 15 Oct 2025, Roubíček et al., 2020)
Dissipative/gradient flows	Staggered auxiliary variable on temporal mesh	(Liu et al., 14 Mar 2025)
High-order finite difference (zigzag)	Explicit staggered grid hybrid schemes	(Poggioni et al., 23 May 2025)

Explicit staggered schemes are the method of choice for modeling sharply heterogeneous, temporally- and spatially-varying flows in gas/hydrogen blending applications (Brodskyi et al., 2024), simulation of large-scale fluctuating systems, operator-splitting for multiscale (diffusive/transport) equations with correct asymptotics (Küpper et al., 2015), and for robust, efficient partitioned algorithms in fluid–structure and electroactive multiphysics (Guo et al., 15 Oct 2025, Seifi et al., 2018). Benchmark studies confirm second-order convergence, exact conservation, and enhanced parallel/algorithmic efficiency relative to monolithic or fully implicit alternatives.

6. Extensions: High-Order and Specialized Staggered Schemes

Recent developments generalize the explicit staggered approach:

• Zigzag schemes: These are explicit staggered-grid finite difference schemes of arbitrary (even infinite) order using a “zigzag” hybrid stencil, whose non-symmetric weights decay as $j^{−1/2}$ in the limit. These schemes exhibit broader stability (CFL) regions than symmetric or upwind stencils, and prevent spurious ghost solutions while remaining easy to implement with legacy codes (Poggioni et al., 23 May 2025).
• Energy-stable auxiliary-variable staggered methods: For dissipative systems, staggered auxiliary variables ensure energy monotonicity (or positiveness) unconditionally, even for arbitrarily large time steps, and admit extensions to systems lacking a known energy lower bound via monotone transforms (such as arctan) (Liu et al., 14 Mar 2025).
• Multi-phase and transport with heterogeneity: Mixture equations on pipeline and networked graphs support partial density and composition staggering, as well as upwinded mixing at junctions for strongly variable composition flows (Brodskyi et al., 2024).
• General mesh and polynomial degrees: The abstract formulation of L²/continuous velocity and discontinuous thermodynamic spaces on unstructured meshes allows arbitrary-order extension (useful in high-fidelity atmospheric or shock hydrodynamics) (Abgrall, 2021).

7. Limitations and Outlook

Explicit staggered schemes, while highly efficient and locally conservative, are fundamentally subject to the CFL conditional stability constraint; simulation of stiff, multiscale, or strongly-coupled fast–slow dynamics may require implicit, semi-implicit, or operator-splitting extensions to be practical for large time steps or highly anisotropic domains (Gyrya et al., 2018, Küpper et al., 2015, Guo et al., 15 Oct 2025). Additionally, the schemes do not capture sharp discontinuities (strong shocks) without further stabilization (artificial dissipation or limiting). In highly nonlinear, non-ideal, or multiphase regimes, physical extension (e.g., real-gas equations of state, variable coefficients) remains possible but may complicate analytic optimized inversion steps at the core of the explicit update formulas (Brodskyi et al., 2024). Nonetheless, explicit staggered schemes remain foundational in large-scale scientific and engineering simulation of hydrodynamics, transport, networked flows, and dissipative systems.

References:

• "An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks" (Gyrya et al., 2018)
• "Simulation of Gas Mixture Dynamics in a Pipeline Network using Explicit Staggered-Grid Discretization" (Brodskyi et al., 2024)
• "Linear, decoupled and positivity-preserving staggered mesh schemes for general dissipative systems with arbitrary energy distributions" (Liu et al., 14 Mar 2025)
• "A Staggered Explicit-Implicit Finite Element Formulation for Electroactive Polymers" (Seifi et al., 2018)
• "Staggered schemes for compressible flow: a general construction" (Abgrall, 2021)
• "Staggered grid leap-frog scheme for the (2+1)D Dirac equation" (Hammer et al., 2013)
• "Staggered Schemes for Fluctuating Hydrodynamics" (Balboa et al., 2011)
• "A Consistent Quasi-Second Order Staggered Scheme for the Two-Dimensional Shallow Water Equations" (Herbin et al., 2021)
• "An Asymptotic Preserving Two-Dimensional Staggered Grid Method for multiscale transport equations" (Küpper et al., 2015)
• "A new class of finite difference methods: The zigzag schemes" (Poggioni et al., 23 May 2025)
• "An Unconditionally Stable Explicit Robin-Robin Partitioned Scheme for Fluid-Structure Interaction" (Guo et al., 15 Oct 2025)
• "Staggered explicit-implicit time-discretization for elastodynamics with dissipative internal variables" (Roubíček et al., 2020)
• "Staggered Scheme for the Compressible Fluctuating Hydrodynamics of Multispecies Fluid Mixtures" (Srivastava et al., 2022)