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

High Order Spectral Volume and Spectral Difference Methods on Unstructured Grids

The spectral volume (SV) and the spectral difference (SD) methods were developed by Wang and Liu and their collaborators for conservation laws on unstructured grids. They were introduced to achieve high-order accuracy in an efficient manner. Recently, these methods were extended to three-dimensional systems and to the Navier Stokes equations. The simplicity and robustness of these methods have made them competitive against other higher order methods such as the discontinuous Galerkin and residual distribution methods. Although explicit TVD Runge-Kutta schemes for the temporal advancement are easy to implement, they suffer from small time step limited by the Courant-Friedrichs-Lewy (CFL) condition. When the polynomial order is high or when the grid is stretched due to complex geometries or boundary layers, the convergence rate of explicit schemes slows down rapidly. Solution strategies to remedy this problem include implicit methods and multigrid methods. A novel implicit lower-upper symmetric Gauss-Seidel (LU-SGS) relaxation method is employed as an iterative smoother. It is compared to the explicit TVD Runge-Kutta smoothers. For some p-multigrid calculations, combining implicit and explicit smoothers for different p-levels is also studied. The multigrid method considered is nonlinear and uses Full Approximation Scheme (FAS). An overall speed-up factor of up to 150 is obtained using a three-level p-multigrid LU-SGS approach in comparison with the single level explicit method for the Euler equations for the 3rd order SD method. A study of viscous flux formulations was carried out for the SV method. Three formulations were used to discretize the viscous fluxes: local discontinuous Galerkin (LDG), a penalty method and the 2nd method of Bassi and Rebay. Fourier analysis revealed some interesting advantages for the penalty method. These were implemented in the Navier Stokes solver. An implicit and p-multigrid method was also implemented for the above. An overall speed-up factor of up to 1500 is obtained using a three-level p-multigrid LU-SGS approach in comparison with the single level explicit method for the Navier-Stokes equations. The SV method was also extended to turbulent flows. The RANS based SA model was used to close the Reynolds stresses. The numerical results are very promising and indicate that the approaches have great potentials for 3D flow problems

Issures in Discontinuous High-Order Methods: Broadband Wave Computation and Viscous Boundary Layer Resolution

A new discontinuous formulation named Correction Procedure via Reconstruction (CPR) was developed for conservation laws. CPR is an efficient nodal differential formulation unifying the discontinuous Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods, is easy to implement. In this thesis, we deal with two issues: the efficient computation of broadband waves, and the proper resolution of a viscous boundary layer with the high-order CPR method. A hybrid discontinuous space including polynomial and Fourier bases is employed in the CPR formulation in order to compute broad-band waves. The polynomial bases are used to achieve a certain order of accuracy, while the Fourier bases are able to exactly resolve waves at a certain frequency. Free-parameters introduced in the Fourier bases are optimized in order to minimize both dispersion and dissipation errors by mimicking the dispersion-relation-preserving (DRP) method for a one-dimensional wave problem. For the one-dimensional wave problem, the dispersion and dissipation properties and the optimization procedure are investigated through a wave propagation analysis. The optimization procedure is verified with a wave propagation analysis and several numerical tests. The two-dimensional wave behavior is investigated through a wave propagation analysis and the wave propagation properties are verified with a numerical test of the two-dimensional acoustic wave equation. In order to understand the mesh size requirement to resolve a viscous boundary layer using CPR method, grid resolution studies are performed. . It is well known that the mesh size, which is defined from non-dimensional wall distance y^+=1, gives accepted results to simulate viscous boundary layer problem for 2nd order finite volume method. For high-order CPR formulation, what grid size y^+ is required to match the results from the 2nd order finite volume method with y^+=1. 1D and 2D burger\u27s equation are used as the viscous boundary layer model problem. Skin friction is used as the indicator of accuracy for the resolution of a boundary layer. Keywords: (Correction Procedure via Reconstruction), A Hybrid Discontinuous Space, Wave Propagation Analysis, Grid Resolution Study, Method of Manufactured Solution

Multigrid solution for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2004.Includes bibliographical references (p. 69-74).A high-order discontinuous Galerkin finite element discretization and p-multigrid solution procedure for the compressible Navier-Stokes equations are presented. The discretization has an element-compact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the p-multigrid solver, which relies on an element-line Jacobi smoother. The element-line Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2-D scalar convection-diffusion shows that the element-line Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h(Ìp+l)). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higher-order is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.by Todd A. Oliver.S.M

Simulation of Multispecies Gas Flows using the Discontinuous Galerkin Method

Truncation errors and computational cost are obstacles that still hinder large-scale applications of the Computational Fluid Dynamics method. The discontinuous Galerkin method is one of the high-order schemes utilized extensively in recent years, which is locally conservative, stable, and high-order accurate. Besides that, it can handle complex geometries and irregular meshes with hanging nodes. In this document, the nondimensional compressible Euler equations and Reynolds- Averaged Navier-Stokes equations are discretized by discontinuous Galerkin methods with a two-equations turbulence model on both structured and unstructured meshes. The traditional equation of state for an ideal gas model is substituted by a multispecies thermodynamics model in order to complete the governing equations. An approximate Riemann solver is used for computing the convective flux, and the diffusive flux is approximated with some internal penalty based schemes. The temporal discretization of the partial differential equations is either performed explicitly with the aid of Rung-Kutta methods or with semi-implicit methods. Inspired by the artificial viscosity diffusion based limiter for shock-capturing method, which has been extensively studied, a novel and robust technique based on the introduction of mass diffusion to the species governing equations to guarantee that the species mass fractions remain positive has been thoroughly investigated. This contact-surface-capturing method is conservative and a high order of accuracy can be maintained for the discontinuous Galerkin method. For each time step of the algorithm, any trouble cell is first caught by the contact-surface discontinuity detector. Then some amount of mass diffusions are added to the governing equations to change the gas mixtures and arrive at an equilibrium point satisfying some conditions. The species properties are reasonable without any oscillations. Computations are performed for many steady and unsteady flow problems. For general non-mixing fluid flows, the classical air-helium shock bubble interaction problem is the central test case for the high-order discontinuous Galerkin method with a mass diffusion based limiter chosen. The computed results are compared with experimental, exact, and empirical data to validate the fluid dynamic solver

Simulation of Multispecies Gas Flows using the Discontinuous Galerkin Method

Truncation errors and computational cost are obstacles that still hinder large-scale applications of the Computational Fluid Dynamics method. The discontinuous Galerkin method is one of the high-order schemes utilized extensively in recent years, which is locally conservative, stable, and high-order accurate. Besides that, it can handle complex geometries and irregular meshes with hanging nodes. In this document, the nondimensional compressible Euler equations and Reynolds- Averaged Navier-Stokes equations are discretized by discontinuous Galerkin methods with a two-equations turbulence model on both structured and unstructured meshes. The traditional equation of state for an ideal gas model is substituted by a multispecies thermodynamics model in order to complete the governing equations. An approximate Riemann solver is used for computing the convective flux, and the diffusive flux is approximated with some internal penalty based schemes. The temporal discretization of the partial differential equations is either performed explicitly with the aid of Rung-Kutta methods or with semi-implicit methods. Inspired by the artificial viscosity diffusion based limiter for shock-capturing method, which has been extensively studied, a novel and robust technique based on the introduction of mass diffusion to the species governing equations to guarantee that the species mass fractions remain positive has been thoroughly investigated. This contact-surface-capturing method is conservative and a high order of accuracy can be maintained for the discontinuous Galerkin method. For each time step of the algorithm, any trouble cell is first caught by the contact-surface discontinuity detector. Then some amount of mass diffusions are added to the governing equations to change the gas mixtures and arrive at an equilibrium point satisfying some conditions. The species properties are reasonable without any oscillations. Computations are performed for many steady and unsteady flow problems. For general non-mixing fluid flows, the classical air-helium shock bubble interaction problem is the central test case for the high-order discontinuous Galerkin method with a mass diffusion based limiter chosen. The computed results are compared with experimental, exact, and empirical data to validate the fluid dynamic solver

Implicit time integration for high-order compressible flow solvers

The application of high-order spectral/hp element discontinuous Galerkin (DG) methods to unsteady compressible flow simulations has gained increasing popularity. However, the time step is seriously restricted when high-order methods are applied to an explicit solver. To eliminate this restriction, an implicit high-order compressible flow solver is developed using DG methods for spatial discretization, diagonally implicit Runge-Kutta methods for temporal discretization, and the Jacobian-free Newton-Krylov method as its nonlinear solver. To accelerate convergence, a block relaxed Jacobi preconditioner is partially matrix-free implementation with a hybrid calculation of analytical and numerical Jacobian.The problem of too many user-defined parameters within the implicit solver is then studied. A systematic framework of adaptive strategies is designed to relax the difficulty of parameter choices. The adaptive time-stepping strategy is based on the observation that in a fixed mesh simulation, when the total error is dominated by the spatial error, further decreasing of temporal error through decreasing the time step cannot help increase accuracy but only slow down the solver. Based on a similar error analysis, an adaptive Newton tolerance is proposed based on the idea that the iterative error should be smaller than the temporal error to guarantee temporal accuracy. An adaptive strategy to update the preconditioner based on the Krylov solver’s convergence state is also discussed. Finally, an adaptive implicit solver is developed that eliminates the need for repeated tests of tunning parameters, whose accuracy and efficiency are verified in various steady/unsteady simulations. An improved shock-capturing strategy is also proposed when the implicit solver is applied to high-speed simulations. Through comparisons among the forms of three popular artificial viscosities, we identify the importance of the density term and add density-related terms on the original bulk-stress based artificial viscosity. To stabilize the simulations involving strong shear layers, we design an extra shearstress based artificial viscosity. The new shock-capturing strategy helps dissipate oscillations at shocks but has negligible dissipation in smooth regions.Open Acces