Entropy Stable Staggered Grid Spectral Collocation for the Burgers' and Compressible Navier-Stokes Equations

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], extends the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to a combination of tensor product Legendre-Gauss (LG) and LGL points. The new semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and the compressible Navier-Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators)

Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.Comment: 43 page

Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions

Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier-Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called super-accuracy is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated. Finally, the influence of the characterizing parameters of the FR method as well as the the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed

High-Order Positivity-Preserving \u3ci\u3eL\u3csub\u3e2\u3c/sub\u3e\u3c/i\u3e-Stable Spectral Collocation Schemes for the 3-D Compressible Navier-Stokes Equations

High-order entropy stable schemes are a popular method used in simulations with the compressible Euler and Navier-Stokes equations. The strength of these methods is that they formally satisfy a discrete entropy inequality which can be used to guarantee L2 stability of the numerical solution. However, a fundamental assumption that is explicitly or implicitly used in all entropy stability proofs available in the literature for the compressible Euler and Navier-Stokes equations is that the thermodynamic variables (e.g., density and temperature) are strictly positive in the entire space{time domain considered. Without this assumption, any entropy stability proof for a numerical scheme solving the compressible Navier-Stokes equations is incomplete. Unfortunately, if the solution loses regularity the positivity assumption may fail to hold for a high-order entropy stable scheme unless special care is taken. To address this problem, we present a new class of positivity-preserving, entropy stable spectral collocation schemes for the 3-D compressible Navier-Stokes equations. The key distinctive property of our method is that it is proven to guarantee the pointwise positivity of density and temperature for compressible viscous flows. The new schemes are constructed by combining a positivity-violating entropy stable method of arbitrary order of accuracy and a novel first-order positivity-preserving entropy stable method discretized on the same Legendre-Gauss-Lobatto (LGL) collocation points used for the high-order counterpart. The proposed framework is general and can be directly extended to other SBP-SAT-type schemes. Numerical results demonstrating accuracy and positivity-preserving properties of the new spectral collocation schemes are presented for viscous and inviscid flows with nearly vacuum regions, very strong shocks, and contact discontinuitie

A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations

In this paper, a compact third-order gas-kinetic scheme is proposed for the compressible Euler and Navier-Stokes equations. The main reason for the feasibility to develop such a high-order scheme with compact stencil, which involves only neighboring cells, is due to the use of a high-order gas evolution model. Besides the evaluation of the time-dependent flux function across a cell interface, the high-order gas evolution model also provides an accurate time-dependent solution of the flow variables at a cell interface. Therefore, the current scheme not only updates the cell averaged conservative flow variables inside each control volume, but also tracks the flow variables at the cell interface at the next time level. As a result, with both cell averaged and cell interface values the high-order reconstruction in the current scheme can be done compactly. Different from using a weak formulation for high-order accuracy in the Discontinuous Galerkin (DG) method, the current scheme is based on the strong solution, where the flow evolution starting from a piecewise discontinuous high-order initial data is precisely followed. The cell interface time-dependent flow variables can be used for the initial data reconstruction at the beginning of next time step. Even with compact stencil, the current scheme has third-order accuracy in the smooth flow regions, and has favorable shock capturing property in the discontinuous regions. Many test cases are used to validate the current scheme. In comparison with many other high-order schemes, the current method avoids the use of Gaussian points for the flux evaluation along the cell interface and the multi-stage Runge-Kutta time stepping technique.Comment: 27 pages, 38 figure

Differential formulation of discontinuous Galerkin and related methods for compressible Euler and Navier-Stokes equations

A new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by the current work is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin (DG), staggered grid, spectral volume (SV), and spectral difference (SD). The approach is then extended to diffusion equation and Navier-Stokes equations. In the discretization of the diffusion terms, the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches are used. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. The current work also includes a study of high-order curve boundaries representations. A new boundary representation based on the Bezier curve is then developed and analyzed, which is shown to have several advantages for complicated geometries. To further enhance the efficiency, the capability of h/p mesh adaptation is developed for the CPR solver. The adaptation is driven by an efficient multi-p a posteriori error estimator. P-adaptation is applied to smooth regions of the flow field while h-adaptation targets the non-smooth regions, identified by accuracy-preserving TVD marker. Several numerical tests are presented to demonstrate the capability of the technique