Higher order and adaptive DG methods for compressible flows

Abstract

In this work, we introduce the consistency and adjoint consistency analysis of discontinuous Galerkin discretizations (Section 2). In particular, we give a general framework for analyzing the consistency and adjoint consistency of DG discretizations for linear problems with inhomogeneous boundary conditions (Section 2.1). This includes the derivation of continuous adjoint problems associated to specific target quantities J(·), the derivation of primal and adjoint residual forms of the discretizations and the discussion whether the discretizations of the primal equations in combination with the discretizations of the target quantities, J_h(·), are adjoint consistent or not. This analysis is then extended to nonlinear problems (Section 2.2). In subsequent sections, we consider DG discretizations of scalar model problems, • the linear advection(-reaction) equation (Section 3), and • Poisson’s equation (Section 4), and of compressible flow problems governed by • the compressible Euler equations (Section 5), • the compressible Navier-Stokes equations (Section 6), and • the Reynolds-averaged Navier-Stokes (RANS) equations and the k-ω turbulence model equations (Section 7). Each of these discretizations will be analyzed with respect to consistency, adjoint consistency, and global and local conservation properties. Furthermore, for the DG discretizations of the scalar model problems we discuss stability properties and give a priori estimates of the discretization error measured in terms of related DG-norms, in terms of the L2-norm as well as in terms of target quantities J(·), (Sections 3 and 4). Furthermore, some of the theoretical results, in particular the a priori error estimates in L2 and J(·), are supported by experimental measurements of the order of convergence (Sections 4.9 and 4.11). For the compressible flow problems, for which no a priori error estimates are available, we provide numerical results and experimental measurements of the order of convergence for inviscid and viscous laminar flows (Sections 5.8 and 6.8). Then, Section 8 is devoted to the derivation of adjoint-based a posteriori estimates of the discretization error in computed target quantity values J_h(u_h) like aerodynamic force coefficients. These estimates are then decomposed into sums of local error indicators (the so-called goal-oriented or adjoint-based indicators) which can be used in an adaptive mesh refinement algorithm tailored to the accurate and efficient approximation of the target quantity. After introduction of this adjoint-based error estimation and mesh refinement approach for single target quantities (Section 8.1), it is extended to the treatment of multiple target quantities (Section 8.2). Section 8.4 finalizes the adaptivity section with the derivation of residual-based indicators. Not depending on an adjoint problem these indicators target at resolving the overall flow field. Finally, in Section 9 the performance of the adjoint-based error estimation, the adjoint-based mesh refinement and the residual-based mesh refinement will be demonstrated for a number of aerodynamic test cases of increasing compexity.