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

TL;DR

The paper tackles the erosion of high-order accuracy in moving-mesh methods for hyperbolic conservation laws when mesh motion is nonsmooth. It introduces transport polynomial exactness (TPE(k)) as a mesh-motion-independent criterion and develops evolved geometric moments (EGMs) to extend the geometric conservation law to higher orders. A key theoretical result shows that second-degree EGMs align with exact geometric moments under SSPRK3 due to a superconvergence effect, enabling a TPE(2) RMM scheme that remains exact for quadratic transport regardless of mesh motion or pseudo-time stepping. Numerical experiments demonstrate quadratic transport accuracy, stable third-order convergence under extreme deformation, and significant efficiency gains by reducing pseudo-time levels, highlighting the scheme’s potential for robust, high-precision moving-mesh simulations.

Abstract

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.

Figures (7)

• Figure 1: Mapping from the reference cell $I_{\mathrm{ref}}$ to the physical cell $I_{i,j}$.
• Figure 1: Mesh configuration for the random rezoning with $|C_r|=0.5$. The mesh undergoes large-scale random perturbations at every physical-time step, resulting in severe distortion and occasional near-triangular cells.
• Figure 2: Geometric evolution of the cell $I_{i,j}(\tau)=A(\tau)B(\tau)C(\tau)D(\tau)$ during a remapping step. The pseudo-time $\tau$ parametrizes the deformation from the current mesh $\mathcal{T}_h^n$ ($\tau=0$) to the updated mesh $\mathcal{T}_h^{n+1}$ ($\tau=\tau^{\mathrm{final}}$).
• Figure 2: Woodward--Colella blast wave problem \ref{['subsubsec:blast_wave']}: density profile along $y=0$ computed by the TPE(2)-ol and Fixed schemes. $N_x\times N_y=400\times 10$, $T=0.038$.
• Figure 3: Shu--Osher problem \ref{['subsubsec:shu_osher']}: density profile along $y=0$ computed by the TPE(2)-ol and Fixed schemes. $N_x\times N_y=400\times 10$, $T=1.8$.

Theorems & Definitions (26)

• Definition 2.1: Transport polynomial exactness
• Definition 3.1: $k$-exact reconstruction
• Remark 3.2
• Lemma 3.3: Linear relation
• Proof 1
• Remark 3.5: Implementation via EGMs
• Remark 3.6
• Proposition 4.1: TPE(2) property of $\mathcal{E}_h$
• Remark 4.2
• Theorem 4.3: Quadratic polynomial preservation under Forward Euler