[Paper Review] A coupled discontinuous Galerkin-Finite Volume framework for solving gas dynamics over embedded geometries

Vincenzo Gulizzi, Ann Almgren+1 more|arXiv (Cornell University)|May 29, 2021

Computational Fluid Dynamics and Aerodynamics90 references15 citations

TL;DR

This paper presents a novel hp-adaptive mesh refinement framework that couples high-order discontinuous Galerkin (dG) and shock-capturing finite volume (FV) schemes on embedded Cartesian grids to solve inviscid gas dynamics with high accuracy and robust shock resolution. By using level-set-based geometry representation and high-order quadrature for curved cut cells, the method achieves high-order accuracy in smooth regions and dynamically tracks shocks via FV at the finest AMR level, demonstrating optimal convergence and accurate discontinuity resolution in 2D and 3D test cases.

ABSTRACT

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.

Motivation & Objective

• To develop a high-order, adaptive numerical framework for solving compressible gas dynamics in complex geometries with embedded boundaries.
• To address the small-cell problem in embedded boundary methods through cell-merging and high-order quadrature on implicitly defined meshes.
• To combine high-order dG schemes for smooth flow regions with shock-capturing FV schemes at the finest AMR level for robust discontinuity resolution.
• To enable dynamic hp-adaptive refinement that optimizes accuracy and efficiency by selecting appropriate order and resolution based on solution features.

Proposed method

• The method uses a block-structured, multi-level Cartesian grid hierarchy with embedded boundaries represented by a level set function.
• Irregular curved elements near boundaries are treated with high-order quadrature rules and a cell-merging strategy to mitigate the small-cell problem.
• High-order dG schemes are applied at coarser AMR levels with variable polynomial orders per level, while the finest level uses a second-order FV scheme for shock capturing.
• The framework employs hp-AMR with dynamic refinement based on solution gradients and discontinuity indicators, using thresholds for density and shock sensors.
• Time integration is performed via high-order TVD Runge-Kutta schemes, ensuring stability and accuracy in explicit time stepping.
• The implementation leverages the AMReX framework for parallel, block-structured AMR, supporting dynamic mesh generation and load balancing.

Experimental results

Research questions

• RQ1 Can a coupled dG-FV approach with hp-AMR achieve high-order accuracy in smooth regions while robustly resolving shocks in embedded geometry problems?
• RQ2 How effectively does cell merging and high-order quadrature mitigate the small-cell problem in embedded boundary methods with curved cut cells?
• RQ3 To what extent can dynamic hp-adaptive refinement balance accuracy and computational cost in complex 2D and 3D gas dynamics simulations?
• RQ4 How does the method perform in capturing complex shock structures, such as Mach reflections and triple points, compared to analytical and experimental benchmarks?

Key findings

• The method achieves optimal convergence rates in smooth flow problems, such as the supersonic vortex test, confirming high-order accuracy of the dG discretization.
• In shock-dominated flows, such as shock reflection from a convex-concave wall, the framework accurately captures the Mach stem and triple point formation, with numerical results closely matching experimental data.
• The computed location of the triple point for Mach numbers M = 1.19, 1.30, and 1.41 aligns well with experimental measurements, validating the shock-capturing capability.
• The use of dynamic hp-AMR with FV at the finest level ensures that shocks are always resolved with sufficient resolution, while smooth regions maintain high-order accuracy.
• The framework successfully handles complex 3D embedded geometries with intricate cut-cell configurations, demonstrating robustness in both static and dynamic AMR settings.
• The method supports variable polynomial orders across AMR levels, enabling efficient resolution of multi-scale flow features without compromising accuracy.