Recovery Discontinuous Galerkin Method for Compressible Turbulence

52nd Aerospace Sciences Meeting

Nair

2014

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Discontinuous Galerkin (DG) methods have seen increased use over the past few decades. Recent work has shown that extremely accurate solutions (3p + 2) can be achieved for pure diffusion problems when using Recovery Discontinuous Galerkin (RDG) methods. However, the order of accuracy for advection is at best 2p+1, thus negating the main advantages of RDG, particularly for advection-diffusion equations, such as the Navier-Stokes equations. In the present paper, we determine a new approach based on monotone upstream centered schemes for conservation laws (MUSCL) to improve the accuracy of advection. Furthermore, we show that RDG is more computationally efficient than certain commonly used diffusion methods, thus making RDG a more viable approach for high-fidelity turbulence simulations. For this purpose, we use three test problems. The interaction of two rarefactions is considered to evaluate convergence of the new advection approach. The problem of decaying isotropic turbulence with eddy shocklets is used to demonstrate the improvement of the proposed approach over standard advection. A pure diffusion Navier-Stokes problem is employed to compare the performance of RDG against other commonly used diffusion schemes.

Section: Numerical Methodologymentioning

confidence: 99%

“…12,13 While the temporal evolution of the kinetic energy and enstrophy were well-resolved on grids of 64 3 , that dilatation was not, particuarly in the more compressible simulations. It is suspected that lower accuracy of the advection scheme is responsible for this phenomenon.…”

Section: Introductionmentioning

confidence: 97%

A Simple Method to Improve the Accuracy of Advection in Discontinuous Galerkin Methods for Navier-Stokes Simulations

52nd Aerospace Sciences Meeting

Nair

2014

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“…3 Compared to other DG modifications for handling parabolic/elliptic PDE behavior, RDG is characterized by extremely high order of accuracy, high computational cost per iteration, and a specialized boundary treatment to maintain high order accuracy. The general concept of Recovery has assisted in the development of similar schemes, for example the reconstruction-based DG method of Luo et al 4 Additionally, Recovery DG has been shown to competent in Direct Numerical Simulation (DNS) of compressible turbulence with periodic boundary conditions by Johnsen et al 5,6 . However, the Recoverybased DG method as developed by Lo and Van Leer, despite its promise, has been sparsely applied to real flow physics problems.…”

Section: Introductionmentioning

confidence: 99%

Progress Towards the Application of the Recovery-Based Disontinuous Galerkin Method to Practical Flow Physics Problems

Johnson

46th AIAA Fluid Dynamics Conference

2016

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Recent progress in the development of the Recovery-Based Discontinuous Galerkin method of Van Leer and Lo is reported. The method is shown to be competent for boundary value problems that involve second order derivatives, including shear terms, such as the viscous terms of the Navier-Stokes equations. As expected, the method's high order of accuracy yields superior results in terms of error versus computational time.

“…2 Compared to other DG approaches for handling parabolic/elliptic PDEs, Recovery DG is characterized by high order of accuracy, small spectral radius, and no ambiguously bounded stabilization parameters. The general concept of recovery has assisted in the development of similar schemes, for example the reconstruction-based DG method of Luo et al, 3 the enhanced stability recovery scheme of Ferrero et al, 4 and the modified recovery scheme (MRDG-1x) of French et al 5 Additionally, Recovery DG has demonstrated competence when applied to the viscous fluxes of the compressible Navier-Stokes equations in Direct Numerical Simulation of compressible turbulence by Johnsen et al 6,7 For a given interface in the computational domain, the recovery process builds a smooth polynomial function (the "recovered" solution) across the union of the neighboring elements, maximizing the use of the solution data associated with each element and providing an unambiguous and accurate approximation for both the solution and its gradient along the interface. The effect of the recovery procedure is demonstrated work will always be referred to as exp(y), and the lowercase e is reserved to refer to element addresses.…”

Section: Introductionmentioning

confidence: 99%

A New Family of Discontinuous Galerkin Schemes for Diffusion Problems

Johnson

23rd AIAA Computational Fluid Dynamics Conference

2017

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Novel Discontinuous Galerkin formulations for parabolic partial differential equations, such as the unsteady diffusion equation, are introduced. Fourier analysis of the various schemes is presented to determine order of accuracy and spectral radius. Additionally, computational examples for the 1D scalar diffusion equation are presented. The method's performance for 2D problems is also demonstrated. Some members of the family exhibit favorable performance in comparison to commonly used methods with regard to accuracy and computational efficiency. Additionally, the method can be adapted to fit within the Flux Reconstruction description of Huynh for diffusion problems; we explore the corresponding 1D formulation. Figure 1: The Recovery process in 2D for a p2 discretization.in Figure 1 for a p2 discretization, taken from our presentation at AIAA Aviation 2016. 8 The recovered solution corresponding to the interface I AB between two neighboring elements Ω A and Ω B , denoted f AB , recovers a very good approximation for a smooth exact solution by applying the recovery procedure to the discontinuous solution representation of U h A and U h B over Ω A ∪ Ω B . The Recovery DG method, extended to multidimensional problems by Lo & Van Leer, 9 shows great promise for purely parabolic PDEs in multiple dimensions, including initial boundary value problems (IBVPs) with time-dependent Dirichlet conditions. 8 However, for combined advection-diffusion problems, where the Recovery DG scheme is employed to handle the diffusive fluxes, it is difficult to design a complete advectiondiffusion DG scheme that captures the advective fluxes with the same accuracy that Recovery DG achieves for the diffusive terms. While the accuracy of the combined method is constrained by the standard DG method's order of accuracy in capturing the advective fluxes, the cost and complication of the method is dominated by the Recovery DG scheme for the diffusive fluxes. This pair of performance bottlenecks is undesirable.With these issues in mind, the aim of our recent studies has been to design new DG schemes for diffusion problems that benefit from the high-order reconstruction provided by the recovery concept while avoiding the complications of Lo & Van Leer's 9 Recovery DG formulation. Our umbrella term for this new family of schemes is "Interface Gradient Recovery," abbreviated IGR. Most of the schemes in this family maintain smaller spectral radii than presently popular DG schemes for diffusion. Simplicity is maintained by starting from the "mixed form" approach for parabolic PDEs, which is familiar to many DG practitioners. Within the mixed form, the IGR family is distinguished by the use of recovery for handling interface terms.The focus of this work is the analysis and performance of the scheme for time-dependent diffusion problems. In section II, the scheme will be described for the multidimensional and 1D cases. Then, we show results from Fourier analysis, providing comparisons in terms of spectral radius and order of accuracy. Next, computationa...