Coupled Discontinuous Galerkin/Finite Difference Solver on Hybrid Meshes for Computational Aeroacoustics

Abstract

T HE field of direct simulation of acoustic waves propagation in the presence of a fluid in motion holds high standards of requirements. This is especially the case when it comes to industrial computational aeroacoustics (CAA), where methods must be able to copewith realistic, complex geometries while involving a high order of accuracy and yet result in affordable solvers in terms of computational resources. In that context, different computational methods have emerged over the years. Resulting from various theoretical backgrounds and approaches, they naturally hold specific characteristics, advantages, and drawbacks. Among them, discontinuous Galerkin (DG)methods [1–4] and finite difference (FD)methods [5– 7] are widely spread. The main characteristics of DG and FD methods are roughly recalled in Table 1. DGmethods arewell adapted to take into account complex geometries as they can deal with unstructured meshes [4,8]. Furthermore, their formulation naturally allows local-order refinement and is well suited for parallel computing. On the other hand, FD methods preferably run on structured meshes and are particularly efficient on Cartesian grids. They are easier to implement and less demanding in terms of CPU resources yet show good numerical dispersion and dissipation properties. Both these methods have been extensively studied, successfully implemented, and applied to that range of problems in industrial configurations. Based on this, a new family of questions have been raised. In particular, the possibility of coupling solvers of different natures in order to locally take advantage of the qualities of each method has already been studied byUtzmann et al. [9–11] in the field of CAA. We will also focus on hybridization techniques of DG/FD schemes based on a computational domain-decomposition approach. A typical motivation behind this is to be able to approximate the solution in the close neighborhood of complex obstacles on an unstructured DG mesh and compute the rest of the field on a Cartesian FD grid in order to alleviate computational time and resources. The purpose of this study is to investigate natural questions raised by such a coupling in order to gather informations on the behavior of the resulting hybrid solver in precision and stability. In this paper, we introduce two two-dimensional (2-D) DG timedomain (DGTD)/FD time-domain (FDTD) hybridization algorithms in the context of the resolution of the linearized Euler equations (LEEs) in two different kinds of geometrical configurations. Our results mainly focus on validation aspects, driven by numerical experiments that were conducted on academic test cases, as theoretical results on the interaction of these two schemes presently seem out of reach in a general case. We also present the behavior of our hybrid solver in the context of an acoustic benchmark problem that consists in the diffraction of an acoustic source by complex-geometries obstacles. The paper is organized as follows. In Sec. II, we present our physical modeling and both the FDTD andDGTD schemes that are used in this study.We also recall their compared properties, advantages, and drawbacks. In Sec. III, after presenting the numerical and theoretical issues raised by the design of a DG/FD coupled solver, we focus on our hybridization strategy and its numerical validation. In Sec. IV, we present an application of our hybrid solver to a complex-geometry acoustic test case, which was proposed at the Fourth CAA Workshop on Benchmark Problems [12].