A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

• Published in: Journal of computational physics Vol. 450; p. 110861
• Main Authors: Gulizzi, Vincenzo, Almgren, Ann S., Bell, John B.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.02.2022; Elsevier Science Ltd
• Subjects: Accuracy; Algorithms; Basis functions; Cartesian coordinates; Computational physics; Discontinuity; Discontinuous Galerkin methods; Domains; Embedded boundaries; Exact solutions; Finite volume method; Finite Volume methods; Galerkin method; Gas dynamics; Grid refinement (mathematics); hp-AMR; Ideal gas; Quadratures; Runge-Kutta method; Shock-capturing schemes; Structured grids (mathematics); Discontinuous Galerkin methods; Shock-capturing schemes; Embedded boundaries; hp-AMR; Finite Volume methods
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110861

We present a computational framework for solving the equations of inviscid gas dynamics using structured grids with embedded geometries. The novelty of the proposed approach is the use of high-order discontinuous Galerkin (dG) schemes and a shock-capturing Finite Volume (FV) scheme coupled via an hp adaptive mesh refinement (hp-AMR) strategy that offers high-order accurate resolution of the embedded geometries. The hp-AMR strategy is based on a multi-level block-structured domain partition in which each level is represented by block-structured Cartesian grids and the embedded geometry is represented implicitly by a level set function. The intersection of the embedded geometry with the grids produces the implicitly-defined mesh that consists of a collection of regular rectangular cells plus a relatively small number of irregular curved elements in the vicinity of the embedded boundaries. High-order quadrature rules for implicitly-defined domains enable high-order accuracy resolution of the curved elements with a cell-merging strategy to address the small-cell problem. The hp-AMR algorithm treats the system with a second-order finite volume scheme at the finest level to dynamically track the evolution of solution discontinuities while using dG schemes at coarser levels to provide high-order accuracy in smooth regions of the flow. On the dG levels, the methodology supports different orders of basis functions on different levels. The space-discretized governing equations are then advanced explicitly in time using high-order Runge-Kutta algorithms. Numerical tests are presented for two-dimensional and three-dimensional problems involving an ideal gas. The results are compared with both analytical solutions and experimental observations and demonstrate that the framework provides high-order accuracy for smooth flows and accurately captures solution discontinuities. •Two- and three-dimensional embedded geometries are resolved with high-order accuracy.•Discontinuous Galerkin (dG) methods provide high-order accuracy in regions of smooth flow.•A Finite Volume (FV) method provides robust shock-tracking capabilities of solution discontinuities.•An hp adaptive mesh refinement strategy enables the coupling between dG schemes and the FV scheme.