A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes
This work provides a rigorous stability guarantee for high-order moving-mesh simulations, addressing a key bottleneck in computational fluid dynamics and related fields.
The authors develop a provably stable discontinuous Galerkin spectral element method for conservation laws on moving hexahedral meshes, using a skew-symmetric ALE formulation. They prove stability, conservation, and free-stream preservation, with numerical verification.
We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results.