Harold L. Atkins

High-accuracy algorithms for computational aeroacoustics

This paper presents an analysis of high-bandwidth operators developed for use with an essentially nonoscillatory (ENO) method. The spatial operators of a sixth-order ENO code are modified to resolve waves with as few as 7 points per wavelength (PPW) by decreasing the formal order of the algorithm. Numerical and analytical solutions are compared for the model problems of plane-wave propagation and sound generation by an oscillating sphere. These problems involve linear propagation, wave steepening, and shock formation. An analysis of the PPW required for sufficient accuracy shows that low-order algorithms need an excessive number of grid points to produce acceptable solutions. In contrast, high-order codes provide good predictions on relatively coarse grids. The high-bandwidth operators produce only modest improvements over the original sixth-order operators for nonlinear problems in which wave steepening is significant ; however, they clearly outperform the original operators for long-distance linear propagation. Because the high-bandwidth operators have the same stencil as the original sixth-order operators, these gains are achieved with no additional computational work.

Analysis of ``p''-Multigrid for Continuous and Discontinuous Finite Element Discretizations

Analysis and numerical experiments examining the behavior and performance of p-multigrid (p = polynomial degree) for solving hp-finite element (FEM) discretizations are presented. We begin by demonstrating the mesh and order independent properties of p-multigrid when used to solve a C0 continuous FEM discretization of the Laplace equation. We then apply pmultigrid to both continuous and discontinuous FEM discretizations of the convection equation. Although 1D Fourier analysis predicts that mesh independent results should be possible for both discretizations, in 2D the results are sensitive to both the mesh resolution and the degree of polynomial approximation. Examining the solutions, we find that for both discretizations, the slowest converging mode is long-wavelength along the streamwise direction and short wavelength normal to this direction. Because of the isotropic coarsening of p-multigrid, this mode is not damped on coarse levels.

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method. 1 https://ntrs.nasa.gov/search.jsp?R=20080014258 2020-06-28T15:20:56+00:00Z

Nonreflective boundary conditions for high-order methods

A different approach to nonreflective boundary conditions for the Euler equations is presented. We are motivated by a need for inflow and outflow boundary conditions that do not limit the useful accuracy of high-order accurate methods. The primary interest is in the propagation and convection of continuous acoustic and convective waves. This new approach employs the exact solution to finite waves to relate interior values and ambient conditions to boundary values. The method is first presented in one dimension and then generalized to multidimensions. Grid refinement studies are used to demonstrate high-order convergence for both one-dimensional and two-dimensional flows

Parallel implementation of the discontinuous Galerkin method

This paper describes a parallel implementation of the discontinuous Galerkin method. Discontinuous Galerkin is a spatially compact method that retains its accuracy and robustness on non-smooth unstructured grids and is well suited for time dependent simulations. Several parallelization approaches are studied and evaluated. The most natural and symmetric of the approaches has been implemented in an object-oriented code used to simulate aeroacoustic scattering. The parallel implementation is MPI-based and has been tested on various parallel platforms such as the SGI Origin, IBM SP2, and clusters of SGI and Sun workstations. The scalability results presented for the SGI Origin show slightly superlinear speedup on a fixed-size problem due to cache effects.

Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method

The efficiency of the quadrature-free form of the dis- continuous Galerkin method in two dimensions, and briefly in three dimensions, is examined. Most of the work for constant-coefficient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the flux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simplified physics and the most efficient element types be used over most of the domain.

Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation

An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection-diffusion problems is presented. The general p-multigrid algorithm for DG discretizations involves a restriction from the p=1 to p=0 discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p-multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1. It is shown that this method is not directly applicable to the convection-diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov-Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1 that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.

On problems associated with grid convergence of functionals

The current use of functionals to evaluate order-of-convergence of a numerical scheme can lead to incorrect values. The problem comes about because of interplay between the errors from the evaluation of the functional, e.g., quadrature error, and from the numerical scheme discretization. Alternative procedures for deducing the order-property of a scheme are presented. The problem is studied within the context of the inviscid supersonic flow over a blunt body; however, the problem and solutions presented are not unique to this example.

Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.

Solving Discontinuous Galerkin Formulations of Poisson's Equation using Geometric and p Multigrid

Methods for solving discontinuous Galerkin formulations of the Poisson equation by coupling p multigrid to geometric multigrid are investigated. The simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p = 0 and then applying geometric multigrid is ineffective. The transition from p = 1 top = 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of the p = 1 discontinuous system are not represented well in thep = 0 space. A new approach is proposed that coarsens from thep = 1 discontinuous space to thep = 1 continuous space. This approach eliminates the problems caused by the p = 1 to p = 0 transition. Furthermore, the p = 1 continuous space is a standard finite element space for which geometric multigrid is well-defined. In addition, when the discontinuous Galerkin equations are restricted to a continuous space, one recovers the Galerkin formulation of continuous finite elements. Thus, applying geometric multigrid to this system is straightforward and effective. Numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid-independent and only weakly sensitive to the polynomial order.