Transport Polynomial Exactness for Direct ALE-DG Methods with Nonsmooth mesh motion: A Conservative Coefficient-Evolution Framework

Abstract

High-order direct arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) methods can suffer severe order reduction under nonsmooth mesh motion despite satisfying the geometric conservation law (GCL). To address this limitation, we propose a new ALE-DG framework applicable to moving triangular and quadrilateral meshes. This framework successfully overcomes the uniform Lipschitz constraint inherent in existing direct ALE-DG methods, retaining high-order accuracy even when the mesh-velocity gradients are unbounded. This capability is driven by transport polynomial exactness (TPE($k$)), which is a structural property stronger than the GCL. TPE($k$) ensures the exact transport of polynomials of degree at most $k$ under arbitrary mesh motion. To incorporate TPE($k$) into a direct ALE-DG method, we introduce a coefficient-evolution (CE) formulation. Reconciling strict local conservation, an indispensable requirement for accurate shock capturing, with the preservation of the TPE(\(k\)) property constitutes the principal technical challenge. We resolve this by evolving the coefficients using the third-order strong-stability-preserving Runge--Kutta method, paired with a conservative correction executed via Simpson's rule integration. Crucially, this correction strictly vanishes for exactly transported quadratic polynomials, thereby ensuring that our third-order scheme rigorously satisfies the TPE($2$) property. Numerical experiments demonstrate that the proposed scheme retains third-order accuracy when the discrete mesh-velocity Lipschitz estimate grows as $\mathcal O(h^{-1})$, whereas a representative existing direct ALE-DG scheme exhibits order reduction under the same mesh motion. Tests with discontinuous solutions further demonstrate robust and high-resolution performance under random and Lagrangian mesh motions.