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NewtonGMRES preconditioning for discontinuous Galerkin discretizations of the NavierStokes equations
 SIAM J. Sci. Comput
, 2008
"... Abstract. We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible NavierStokes equations. The spatial discretization is carried out using a Discontinuous Galerkin method with fourth order polynomial interpolations on tri ..."
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Cited by 39 (11 self)
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Abstract. We study preconditioners for the iterative solution of the linear systems arising in the implicit time integration of the compressible NavierStokes equations. The spatial discretization is carried out using a Discontinuous Galerkin method with fourth order polynomial interpolations on triangular elements. The time integration is based on backward difference formulas resulting in a nonlinear system of equations which is solved at each timestep. This is accomplished using Newton’s method. The resulting linear systems are solved using a preconditioned GMRES iterative algorithm. We consider several existing preconditioners such as blockJacobi and GaussSeidel combined with multilevel schemes which have been developed and tested for specific applications. While our results are consistent with the claims reported, we find that these preconditioners lack robustness when used in more challenging situations involving low Mach numbers, stretched grids or high Reynolds number turbulent flows. We propose a preconditioner based on a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fillin (ILU0) of the Jacobian matrix. The performance of the ILU0 smoother is found to depend critically on the element numbering. We propose a numbering strategy based on minimizing the discarded fillin in a greedy fashion. The coarse scale correction scheme is found to be important for diffusion dominated
A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
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An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method
, 2005
"... Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowsk ..."
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Cited by 29 (0 self)
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Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................
A highorder discontinuous Galerkin multigrid solver for . . .
, 2004
"... Results are presented from the development of a highorder discontinuous Galerkin finite element solver using pmultigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and pmultigrid is outlined for the nonlinear Euler equations of gas dynamics. Anal ..."
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Cited by 18 (2 self)
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Results are presented from the development of a highorder discontinuous Galerkin finite element solver using pmultigrid with line Jacobi smoothing. The line smoothing algorithm is presented for unstructured meshes, and pmultigrid is outlined for the nonlinear Euler equations of gas dynamics. Analysis of 2D advection shows the improved performance of line implicit versus block implicit relaxation. Through a mesh refinement study, the accuracy of the discretization is determined to be the optimal O(h p+1) for smooth problems in 2D and 3D. The multigrid convergence rate is found to be independent of the interpolation order but weakly dependent on the grid size. Timing studies for each problem indicate that higher order is advantageous over grid refinement when high accuracy is required. Finally, parallel versions of the 2D and 3D solvers demonstrate close to ideal coarsegrain scalability.
Shock Capturing with PDEBased Artificial Viscosity for DGFEM: Part I, Formulation
"... Artificial viscosity can be combined with a higherorder discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a nonsmooth artificial viscosity model is employed with an otherwise higherorder approximation, elementtoelement variations in ..."
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Cited by 14 (5 self)
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Artificial viscosity can be combined with a higherorder discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a nonsmooth artificial viscosity model is employed with an otherwise higherorder approximation, elementtoelement variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a higherorder, statebased artificial viscosity with an associated governing partial differential equation (PDE). In the governing PDE, a shock indicator acts as a forcing term while gridbased diffusion is added to smooth the resulting artificial viscosity. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDEbased artificial viscosity is less susceptible to errors introduced by grid edges oblique to captured shocks and boundary layers, thereby enabling accurate heat transfer predictions.
Facilitating the adoption of unstructured highorder methods amongst a wider community of fluid dynamicists
 Mathematical Modelling of Natural Phenomena
"... Abstract. Theoretical studies and numerical experiments suggest that unstructured highorder methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing highorder schemes are generally less robust and more complex ..."
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Cited by 12 (5 self)
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Abstract. Theoretical studies and numerical experiments suggest that unstructured highorder methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing highorder schemes are generally less robust and more complex to implement than their loworder counterparts. These issues, in conjunction with difficulties generating highorder meshes, have limited the adoption of highorder techniques in both academia (where the use of loworder schemes remains widespread) and industry (where the use of loworder schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of highorder methods amongst a nonspecialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured highorder meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and finally the development of highorder methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known highorder schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework.
An efficient low memory implicit dg algorithm for time dependent problems
 Proceedings of the 44th AIAA Aerospace Sciences Meeting
"... We present an efficient implicit time stepping method for Discontinuous Galerkin discretizations of the compressible NavierStokes equations on unstructured meshes. The Local Discontinuous Galerkin method is used for the discretization of the viscous terms. For unstructured meshes, the Local Discont ..."
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Cited by 12 (6 self)
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We present an efficient implicit time stepping method for Discontinuous Galerkin discretizations of the compressible NavierStokes equations on unstructured meshes. The Local Discontinuous Galerkin method is used for the discretization of the viscous terms. For unstructured meshes, the Local Discontinuous Galerkin method is known to produce noncompact discretizations. In order to circumvent the difficulties accociated with this noncompactness, we represent the irregular matrices arising from the discretization algorithm as a product of matrices with a more structured pattern. Time integration is carried out using backward difference formulas. This leads to a nonlinear system of equations to be solved at each timestep. In this paper, we study various iterative solvers for the linear systems of equations that arise in the Newton algorithm. We show that a twolevel preconditioner with incomplete LU as a presmoother is highly efficient yet inexpensive to compute and to store. It performs particularly well for low Mach number flows, where it is more than a magnitude more efficient than pure twolevel or ILU preconditioning. Our methods are demonstrated using three typical test problems with various parameters and timesteps. I.
Scalable Parallel NewtonKrylov Solvers for Discontinuous Galerkin discretizations.” AIAA
, 2009
"... We present techniques for implicit solution of discontinuous Galerkin discretizations of the NavierStokes equations on parallel computers. While a blockJacobi method is simple and straightforward to parallelize, its convergence properties are poor except for simple problems. Therefore, we conside ..."
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Cited by 11 (4 self)
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We present techniques for implicit solution of discontinuous Galerkin discretizations of the NavierStokes equations on parallel computers. While a blockJacobi method is simple and straightforward to parallelize, its convergence properties are poor except for simple problems. Therefore, we consider NewtonGMRES methods preconditioned with blockincomplete LU factorizations, with optimized element orderings based on a minimum discarded fill (MDF) approach. We discuss the difficulties with the parallelization of these methods, but also show that with a simple domain decomposition approach, most of the advantages of the blockILU over the blockJacobi preconditioner are still retained. The convergence is further improved by incorporating the matrix connectivities into the mesh partitioning process, which aims at minimizing the errors introduced from separating the partitions. We demonstrate the performance of the schemes for realistic two and threedimensional flow problems. I.
An Adaptive Simplex CutCell Method for Discontinuous Galerkin
 Discretizations of the NavierStokes Equations,” AIAA Paper
, 2007
"... A cutcell adaptive method is presented for highorder discontinuous Galerkin discretizations in two and three dimensions. The computational mesh is constructed by cutting a curved geometry out of a simplex background mesh that does not conform to the geometry boundary. The geometry is represented w ..."
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Cited by 11 (6 self)
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A cutcell adaptive method is presented for highorder discontinuous Galerkin discretizations in two and three dimensions. The computational mesh is constructed by cutting a curved geometry out of a simplex background mesh that does not conform to the geometry boundary. The geometry is represented with cubic splines in two dimensions and with a tesselation of quadratic patches in three dimensions. Highorder integration rules are derived for the arbitrarilyshaped areas and volumes that result from the cutting. These rules take the form of quadraturelike points and weights that are calculated in a preprocessing step. Accuracy of the cutcell method is verified in both two and three dimensions by comparison to boundaryconforming cases. The cutcell method is also tested in the context of outputbased adaptation, in which an adjoint problem is solved to estimate the error in an engineering output. Twodimensional adaptive results for the compressible NavierStokes equations illustrate automated anisotropic adaptation made possible by triangular cutcell meshing. In three dimensions, adaptive results for the compressible Euler equations using isotropic refinement demonstrate the feasibility of automated meshing with tetrahedral cut cells and a curved geometry representation. In addition, both the two and threedimensional results indicate that, for the cases tested, p = 2 and p = 3 solution approximation achieves the userprescribed error tolerance more efficiently compared to p = 1 and p = 0. I.
A Linear Multigrid Preconditioner for the solution of the NavierStokes Equations using a Discontinuous Galerkin Discretization
, 2007
"... A NewtonKrylov method is developed for the solution of the steady compressible NavierStokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element LineJacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum couplin ..."
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Cited by 11 (2 self)
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A NewtonKrylov method is developed for the solution of the steady compressible NavierStokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element LineJacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete blockLU factorization (BlockILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling used for the element LineJacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, Oneway Dissection, Quotient Minimum Degree, Reverse CuthillMckee) especially for viscous test cases. The BlockILU(0) factorization is performed inplace and a novel algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. A linear pmultigrid algorithm using element LineJacobi, and BlockILU(0) smoothing is presented as a preconditioner to GMRES. The coarse level Jacobians are obtained using a