A Parallel Implementation of the Discontinuous-Galerkin-Hancock Method on Unstructured Meshes with Adaptive Mesh Refinement

Abstract

The steady increase in computational power over the last generation gives rise to fields of research that use it, which is why Computational Fluid Dynamics (CFD) has been, and remains, a rapidly evolving field of research. Traditionally, the Navier-Stokes (NS) equations have been used almost exclusively to model viscous gas flows. More recently, the field of moment methods have emerged as an alternative. Moment methods are derived from the kinetic theory of gases. Because of this, they offer certain advantages over the NS model, such as an improved accuracy in describing rarefied gases or multiphase flow, where the microscopic scale can be more prominent. These moment methods are described by first-order hyperbolic partial differential equations (PDE) with stiff local relaxation source terms. The discontinuous-Galerkin (DG) methods, initially designed to solve first order hyperbolic balance laws, are of particular interest in the current study. The discontinuous-Galerkin-Hancock (DGH) method is of third-order accuracy, was designed specifically for the efficient solution of moment methods, and lends itself well to large-scale parallel computing. Once a scheme like DGH has been identified, it is always coupled with a tessellation of space. In a structured discretization, space is often split into quadrilaterals, which can be numbered in an orderly manner, such as would be considered intuitive or “structured”. Contrarily, an unstructured discretization splits space into polygons of theoretically any number of sides. This allows complex geometries to be more easily represented, and hence is more versatile. Given these meshes’ unstructured nature, their implementation becomes less straightforward. The current study outlines a paralellized implementation of the DGH scheme on unstructured meshes in two and three dimensions. An adaptive refinement algorithm was developed alongside this. The implementation allows the computational tasks to be spread amongst many processors, and its adaptive nature allows this implementation to tailor its mesh’s detail to the solution that is being computed in real-time. A multitude of different first-order PDEs in different contexts are solved in the current study. To begin with, a supersonic flow-past-bump case is investigated using the compressible Euler equations. This is done to showcase the adaptive refinement algorithm’s power, combined with the scheme’s ability to capture strong shocks. Next, the unstructured nature of the implementation is taken advantage of with complex geometries, such as for a multi-element airfoil, and for a room equipped with sound diffusion. The flow around the airfoil is modeled using the 10-moment Gaussian moment closure, to capture viscosity’s effects with a first-order hyperbolic model. Finally, the linear convection equations are used to verify the scheme’s order of accuracy on 2D and 3D meshes, and the Euler equations are used on a varying number of CPUs to demonstrate the implementation’s scalability on large clusters.