Georg May

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.

An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow

During the past decade gas-kinetic methods based on the BGK simplification of the Boltzmann equation have been employed to compute fluid flow in a finite-difference or finite-volume context. Among the most successful formulations is the finite-volume scheme proposed by Xu [K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (48) (2001) 289-335]. In this paper we build on this theoretical framework mainly with the aim to improve the efficiency and convergence of the scheme, and extend the range of application to three-dimensional complex geometries using general unstructured meshes. To that end we propose a modified BGK finite-volume scheme, which significantly reduces the computational cost, and improves the behavior on stretched unstructured meshes. Furthermore, a modified data reconstruction procedure is presented to remove the known problem that the Chapman-Enskog expansion of the BGK equation fixes the Prandtl number at unity. The new Prandtl number correction operates at the level of the partial differential equations and is also significantly cheaper for general formulations than previously published methods. We address the issue of convergence acceleration by applying multigrid techniques to the kinetic discretization. The proposed modifications and convergence acceleration help make large-scale computations feasible at a cost competitive with conventional discretization techniques, while still exploiting the advantages of the gas-kinetic discretization, such as computing full viscous fluxes for finite volume schemes on a simple two-point stencil.

A Spectral Difference Method for the Euler and Navier-Stokes Equations on Unstructured Meshes

This work focuses on the extension of the recently introduced Spectral Dierence Method to viscous flow. The spectral dierence method is a conservative pseudo-spectral scheme based on a local collocation on unstructured elements. Recently results for scalar transport equations and the Euler equations have been presented. For the extension to viscous flow several techniques are investigated, such as a central discretization and a split upwind/downwind discretization, akin to the procedure used in the LDG method.

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

Summary We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright

Unstructured Algorithms for Inviscid and Viscous Flows Embedded in a Unified Solver Architecture: Flo3xx

This paper reports recent progress towards a new platform for computational aerodynamics on arbitrary meshes, tentatively designated Flo3xx. This tool is designed for maximum flixibility to serve as both an industrial strength flow solver on general grids, and as a framework for advanced research in the area of CFD and aerodynamic design. Such a flexible platform is crucial for transfering new research to industrial applications. We describe the capabilities and methods used in this tool, and show results from initial validation on inviscid and viscous testcases. Furthermore some open issues in CFD on unstructured grids will be addressed, such as viscous discretization and multigrid methods in an unstructured context.

A High-Order Discontinuous Galerkin Discretization with Multiwavelet-Based Grid Adaptation for Compressible Flows

Multiresolution-based mesh adaptivity using biorthogonal wavelets has been quite successful with finite volume solvers for compressible fluid flow. The extension of the multiresolution-based mesh adaptation concept to high-order discontinuous Galerkin discretization can be performed using multiwavelets, which allow for higher-order vanishing moments, while maintaining local support. An implementation for scalar one-dimensional conservation laws has already been developed and tested. In the present paper we extend this strategy to systems of equations, in particular to the equations governing inviscid compressible flow.

High-Order Accurate Methods for High-Speed Flow

This work focuses on high-order numerical schemes for conservation laws with emphasis on problems that admit discontinuous solutions. We will investigate several enabling techniques with the aim to construct robust methods for unstructured meshes capable of reliably producing globally high-order accurate results even in the presence of discontinuities. We use spectral and pseudo-spectral schemes in connection with such techniques as Gibbs-complementary reconstruction to recover high-order accuracy near discontinuities. I. Introduction Computational aerodynamics has been dominated by schemes which are higher than first-order accurate only in smooth regions, where they are typically restricted to second, or perhaps third order accuracy. The presence of discontinuities in transonic aerodynamics has greatly hampered the success of high-order accurate methods. In the vicinity of discontinuities the typical scheme uses restrictive limiters to damp out high-frequency oscillatory modes. While low-order schemes have been very successful due to their robustness and relative eciency, many fields of research, such as aeroacoustics, large-eddy simulation (LES), and unsteady nonperiodic flow require highly accurate numerical methods. For many applications a high-order accurate scheme for solutions with discontinuities that retains its nominal accuracy everywhere would be quite welcome. In this paper we will explore one possible route to such a scheme. We will start from a most promising new numerical scheme for unstructured meshes, namely the Spectral Dierence method,

A hybrid multilevel method for high-order discretization of the Euler equations on unstructured meshes

Higher order discretization has not been widely successful in industrial applications to compressible flow simulation. Among several reasons for this, one may identify the lack of tailor-suited, best-practice relaxation techniques that compare favorably to highly tuned lower order methods, such as finite-volume schemes. In this paper we investigate solution algorithms in conjunction with high-order Spectral Difference discretization for the Euler equations, using such techniques as multigrid and matrix-free implicit relaxation methods. In particular we present a novel hybrid multilevel relaxation method that combines (optionally matrix-free) implicit relaxation techniques with explicit multistage smoothing using geometric multigrid. Furthermore, we discuss efficient implementation of these concepts using such tools as automatic differentiation.

On the Convergence of a Shock Capturing Discontinuous Galerkin Method for Nonlinear Hyperbolic Systems of Conservation Laws

In this paper, we present a shock capturing discontinuous Galerkin method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [A. Hiltebrand and S. Mishra, Numer. Math., 126 (2014), pp. 103--151], our approach is new in that we do not use streamline diffusion stabilization. It is proved that an artificial viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued solution. The result is valid for general systems and for the arbitrary order discontinuous Galerkin method.

A Hybridized Discontinuous Galerkin Method for Three-Dimensional Compressible Flow Problems

We present a hybridized discontinuous Galerkin method for three-dimensional flow problems. As an implementation technique hybridization is a classic paradigm for dual-mixed finite element discretizations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the numerical mesh. Solving for these thus involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The accuracy of the method has been validated with a scalar convection-diffusion test case. Results are shown for external, compressible flow.