Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes

TL;DR

This paper develops a robust, high-order entropy-stable discontinuous Galerkin spectral element method for complex, curvilinear meshes, enabling stable simulations of hyperbolic-parabolic PDEs on diverse element shapes.

Contribution

It extends entropy-stable DGSEM to arbitrary element shapes using coordinate transformations and a memory-efficient approach, broadening applicability to complex geometries.

Findings

Method achieves entropy stability and convergence on complex meshes.

Successfully applied to flow around a research model.

Maintains stability for hyperbolic-parabolic PDEs on hybrid meshes.

Abstract

Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical approximation. However, high-order methods are prone to instabilities in the presence of underresolved flow features. A popular counter measure to stabilize DG methods is the use of entropy-stable formulations based on summation-by-parts (SBP) operators. The present paper aims to construct a robust and efficient entropy-stable discontinuous Galerkin spectral element method (DGSEM) of arbitrary order on heterogeneous, curvilinear grids composed of triangular and quadrilateral elements or hexahedral, prismatic, tetrahedral and pyramid elements. To the author's knowledge, with the exception of hexahedral and quadrilateral elements, entropy-stable DGSE operators have been constructed exclusively for tetrahedral and triangular meshes. The extension of the DGSEM to more complex element shapes is achieved by means of a collapsed coordinate transformation. Legendre--Gauss quadrature nodes are employed as collocation points in conjunction with a generalized SBP operator and entropy-projected variables. The purely hyperbolic operator is extended to hyperbolic-parabolic problems by the use of a lifting procedure. To circumvent the penalizing time step restriction imposed by the collapsing, modal rather than nodal degrees of freedom are evolved in time, thereby relying on a memory-efficient weight-adjusted approximation to the inverse of the mass matrix. Essential properties of the proposed numerical scheme including free-stream preservation, polynomial and grid convergence as well as entropy conservation / stability are verified. Finally, with the flow around the common research model, the applicability of the presented method to real-world problems is demonstrated.