Beyond Free-Stream Preservation: Transport Polynomial Exactness for Moving-Mesh Methods under Arbitrary Mesh Motion

High-order moving-mesh methods can effectively reduce numerical diffusion, but their formal accuracy typically relies on the regularity of the mesh velocity. This dependency creates a fundamental conflict in the numerical solution of hyperbolic conservation laws, where solution-driven adaptation may induce nonsmooth mesh motion, thereby degrading convergence order. We introduce \emph{transport polynomial exactness} (TPE($k$)), a mesh-motion-independent criterion that generalizes classical free-stream preservation (TPE(0)) to the exact advection of degree-$k$ polynomials. We show that the classical geometric conservation law (GCL) is insufficient to ensure TPE($k$) for $k \ge 1$ due to mismatches in higher-order geometric moments. To resolve this, we propose \emph{evolved geometric moments} (EGMs), obtained by solving auxiliary transport equations discretized compatibly with the physical variables. We rigorously prove that second-degree EGMs evolved via the third-order strong stability preserving Runge–Kutta (SSPRK3) method coincide with the exact geometric moments. This exactness arises from a \emph{superconvergence} mechanism wherein SSPRK3 reduces to Simpson’s rule for EGM evolution. Leveraging this result, we construct a third-order conservative finite-volume rezoning moving-mesh scheme. The scheme satisfies the TPE(2) property for \emph{arbitrary mesh motion} and \emph{any pseudo-time step size}, thereby naturally accommodating spatiotemporally discontinuous mesh velocity. Crucially, this \emph{breaks the efficiency bottleneck} in the conventional advection-based remapping step and reduces the required pseudo-time levels from $\mathcal{O}(h^{-1})$ to $\mathcal{O}(1)$ under bounded but discontinuous mesh velocity. Numerical experiments verify exact quadratic transport and stable third-order convergence under extreme mesh deformation, demonstrating substantial efficiency gains.

💡 Research Summary

The paper addresses a fundamental limitation of high‑order moving‑mesh (rezoning) methods: their formal accuracy traditionally depends on the smoothness of the mesh velocity. When mesh motion is driven by solution features, the velocity can become discontinuous, causing a loss of convergence order. To overcome this, the authors introduce Transport Polynomial Exactness (TPE(k)), a mesh‑motion‑independent criterion that generalizes the classical free‑stream preservation (TPE(0)) to the exact advection of degree‑k polynomials. A scheme is said to be TPE(k) if, for the linear advection equation with a polynomial initial condition of degree k, the fully discrete solution coincides with the exact translated polynomial at all discrete times.

The paper shows that the traditional Geometric Conservation Law (GCL) only guarantees TPE(0) because it enforces compatibility for cell volume alone. Higher‑order geometric moments (integrals of x^s y^r over each cell) are required to preserve the polynomial structure for k ≥ 1. To enforce these moments, the authors propose Evolved Geometric Moments (EGMs): auxiliary transport equations for each moment, discretized with the same numerical scheme used for the physical variables.

A key theoretical result is that when the third‑order Strong Stability Preserving Runge–Kutta (SSPRK3) method is used to integrate the EGMs, the second‑degree moments evolve exactly, i.e., the numerical update coincides with the exact geometric moments. This is proved by demonstrating that SSPRK3 reduces to Simpson’s rule for the moment equations, a phenomenon the authors call superconvergence.

Leveraging exact EGMs, the authors construct a third‑order conservative finite‑volume rezoning moving‑mesh (RMM) scheme that satisfies TPE(2) for arbitrary mesh motion and any pseudo‑time step size. The scheme consists of three operators:

1. Physical evolution (E_h) – a high‑order fixed‑mesh solver (e.g., WENO/DG) applied over the physical time step.
2. Rezoning (Z_h) – a mesh‑adaptation step that may be based on equidistribution, variational principles, or spring analogies.
3. Remapping (R_h) – a pseudo‑time transport ∂_τ U = 0 solved with SSPR​K3, using a 2‑exact hybrid WENO reconstruction together with the EGMs.

Because the TPE(2) property holds independently of the pseudo‑time step Δτ, the remapping requires only O(1) pseudo‑time levels, even when the mesh velocity is bounded but discontinuous. This dramatically reduces the computational bottleneck of traditional advection‑based remapping, which typically needs O(h⁻¹) pseudo‑time steps to maintain high‑order accuracy.

The authors validate the approach with several numerical experiments:

• Exact quadratic transport: a 2‑D linear advection problem with a quadratic polynomial initial condition demonstrates that the scheme transports the polynomial exactly, confirming TPE(2).
• Extreme mesh deformation: meshes undergoing severe stretching, folding, and discontinuous velocity fields still achieve third‑order convergence for smooth solutions.
• Efficiency comparison: the new scheme attains the same accuracy as conventional methods while using far fewer pseudo‑time stages, confirming the O(1) claim.

The paper also discusses implementation advantages: the EGMs eliminate the need for repeated reference‑to‑physical mappings and Jacobian evaluations, making the method “mapping‑ and Jacobian‑free”. Moreover, the framework is compatible with existing high‑order fixed‑grid solvers, allowing straightforward integration into legacy codes.

In summary, the work makes four major contributions:

1. Formalization of TPE(k) as a rigorous, mesh‑independent accuracy metric.
2. Extension of GCL via EGMs, ensuring higher‑order geometric consistency.
3. Proof of exact second‑degree moment evolution under SSPRK3, revealing a superconvergence mechanism.
4. Construction of a TPE(2) RMM scheme that achieves third‑order accuracy with only O(1) pseudo‑time steps, breaking the efficiency bottleneck of advection‑based remapping.

These results open the door to robust, high‑order moving‑mesh simulations for hyperbolic conservation laws, even in the presence of solution‑driven, discontinuous mesh motions, and suggest promising extensions to higher‑order TPE(k), nonlinear limiters, and multi‑physics applications.