A discussion of convergence acceleration techniques as they relate to computational fluid dynamics problems on unstructured meshes is given. Rather than providing a detailed description of particular methods, the various different building blocks of current solution techniques are discussed and examples of solution strategies using one or several of these ideas are given. Issues relating to unstructured grid CFD problems are given additional consideration, including suitability of algorithms to current hardware trends, memory and cpu tradeoffs, treatment of non-linearities, and the development of efficient strategies for handling anisotropy-induced stiffness. The outlook for future potential improvements is also discussed.

A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(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 Line-Jacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place 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 p-multigrid algorithm using element Line-Jacobi, and Block-ILU(0) smoothing is presented as a preconditioner to GMRES. The coarse level Jacobians are obtained using a

A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place 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 p-multigrid algorithm using element Line-Jacobi and Block-ILU(0) smoothing is presented as a preconditioner to GMRES. The linear multigrid preconditioner is shown to significantly improve convergence in terms of the number of linear iterations as well as to reduce the total CPU time required to obtain a converged solution. I.

Introduction D ESPITE impressive progress in both algorithms and computers, the time required to compute the solution of the steady compressible Navier-Stokes equations for the flow about aerodynamic configurations remains excessive for routine use in aircraft design. Consequently, numerical solution techniques applicable to simpler physical models, such as panel methods or solvers incorporating the boundary-layer equations, are heavily used in the design process, despite their limitations. Although a substantial portion of the time required in a simulation based on the Navier-Stokes equations is involved in problem setup, including geometry definition and grid generation, and this needs to be addressed, the solution time is also excessive. One way to reduce the solution time is to improve the iterative method used to achieve a steady state. 1 Alternatively, the accuracy of the spatial discretization can be increased. This allows a reduction in the n

A parallel implicit adaptive mesh refinement (AMR) algorithm is described for the system of partial-differential equations governing steady two-dimensional compressible gaseous flows. The AMR algorithm uses an upwind finite-volume spatial discretization procedure in conjunction with limited linear solution reconstruction and Riemann-solver based flux functions to solve the governing equations on multi-block mesh composed of structured curvilinear blocks with quadrilateral computational cells. A flexible block-based hierarchical data structure is used to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. A matrix-free inexact Newton method is used to solve the system of nonlinear equations arising from this finite-volume spatial discretization procedure and a preconditioned generalized minimal residual (GMRES) method is used to solve the resulting non-symmetric system of linear equations at each step of the Newton algorithm. Right preconditioning of the linear system is used to improve performance of the Krylov subspace method. An additive Schwarz global preconditioner with variable overlap is used in conjunction with block-fill incomplete lower-upper (BFILU) type preconditioners based on the Jacobian of the first-order upwind scheme for each sub-domain. The Schwarz preconditioning and block-based data structure readily allow efficient and scalable parallel implementations of the implicit AMR approach on distributed-memory multi-processor architectures. Numerical results are described for several flow cases, demonstrating both the effectiveness of the mesh adaptation and algorithm parallel performance. The proposed parallel implicit AMR method allows for anisotropic mesh refinement and appears to be well suited for predicting complex flows with disparate spatial and temporal scales in a reliable and efficient fashion. I.

In this paper, we present and justify the strategies and parameters used in PROBE, a Newton-Krylov solver for steady aerodynamic flows. The Krylov solver GMRES is used in matrix-free form. An approximate Jacobian matrix with some modifications to increase the magnitude of the diagonal entries is used to form a preconditioner based on an incomplete lower-upper factorization with two levels of fill. The resulting solver is very efficient, generally reducing the residual twelve orders of magnitude with a CPU expense equivalent to between 500 and 1000 function evaluations. For all cases studied, PROBE converged significantly faster than an approximately-factored multigrid algorithm used for comparison. We propose a convergence rate based on the reduction in the residual obtained in the CPU time required for one function evaluation. With this definition, PROBE achieves a convergence rate per function evaluation between 0.945 and 0.972 for the cases studied, which include inviscid, laminar, and turbulent flows. This is substantially faster than many current algorithms applied to similar flows.

A fast Newton–Krylov algorithm is presented that solves the turbulent Navier–Stokes equations on unstructured 2-D grids. The model of Spalart and Allmaras provides the turbulent viscosity and is loosely coupled to the mean-flow equations. It is often assumed that the turbulence model must be fully coupled to obtain the full benefit of an inexact Newton algorithm. We demonstrate that a loosely coupled algorithm is effective and has some advantages, such as reduced storage requirements and smoother transient oscillations. A transonic single-element case converges to 1 10 12 in 90 s on recent commodity hardware, whereas the lift coefficient is converged to three figures in one quarter of that time. Nomenclature c = reference chord length e = total internal energy F = the inviscid flux tensor G = the stress and heat flux tensor ^i = unit normal aligned with reference chord ^j = unit normal perpendicular to reference chord M1 = freestream Mach number ^n = generic unit normal vector

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Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an efficient high-fidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higher-order hybridizable