The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids

Novel limiters based on the weighted average procedure are developed for finite volume methods solving multi-dimensional hyperbolic conservation laws on unstructured grids. The development of these limiters is inspired by the biased averaging procedure of Choi and Liu [10]. The remarkable features of the present limiters are the new biased functions and the weighted average procedure, which enable the present limiter to capture strong shock waves and achieve excellent convergence for steady state computations. The mechanism of the developed limiters for eliminating spurious oscillations in the vicinity of discontinuities is revealed by studying the asymptotic behavior of the limiters. Numerical experiments for a variety of test cases are presented to demonstrate the superior performance of the proposed limiters.     

Unsteady CFD computations using vertex‐centered finite volumes for unstructured grids on Graphics Processing Units

This paper presents a Navier–Stokes solver for steady and unsteady turbulent flows on unstructured/hybrid grids, with triangular and quadrilateral elements, which was implemented to run on Graphics Processing Units (GPUs). The paper focuses on programming issues for efficiently porting the CPU code to the GPU, using the CUDA language. Compared with cell‐centered schemes, the use of a vertex‐centered finite volume scheme on unstructured grids increases the programming complexity since the number of nodes connected by edge to any other node might vary a lot. Thus, delicate GPU memory handling is absolutely necessary in order to maximize the speed‐up of the GPU implementation with respect to the Fortran code running on a single CPU core. The developed GPU‐enabled code is used to numerically study steady and unsteady flows around the supercritical airfoil OAT15A, by laying emphasis on the transonic buffet phenomenon. The computations were carried out on NVIDIA's Ge‐Force GTX 285 graphics cards and speed‐ups up to ～46 × (on a single GPU, with double precision arithmetic) are reported. Copyright © 2010 John Wiley & Sons, Ltd.     

High‐order k‐exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids

This paper presents a family of High‐order finite volume schemes applicable on unstructured grids. The k‐exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock‐capturing capability of the scheme. This procedure combines the simplicity of the k‐exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High‐order accurate shock capturing schemes. To further improve the efficiency, an efficient problem‐independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non‐oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High‐order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

Multigrid convergence acceleration for implicit and explicit solution of Euler equations on unstructured grids

The multigrid method is one of the most efficient techniques for convergence acceleration of iterative methods. In this method, a grid coarsening algorithm is required. Here, an agglomeration scheme is introduced, which is applicable in both cell‐center and cell‐vertex 2 and 3D discretizations. A new implicit formulation is presented, which results in better computation efficiency, when added to the multigrid scheme. A few simple procedures are also proposed and applied to provide even higher convergence acceleration. The Euler equations are solved on an unstructured grid around standard transonic configurations to validate the algorithm and to assess its superiority to conventional explicit agglomeration schemes. The scheme is applied to 2 and 3D test cases using both cell‐center and cell‐vertex discretizations. Copyright © 2009 John Wiley & Sons, Ltd.     

Spatially adaptive sparse grids for high-dimensional data-driven problems

Sparse grids allow one to employ grid-based discretization methods in data-driven problems. We present an extension of the classical sparse grid approach that allows us to tackle high-dimensional problems by spatially adaptive refinement, modified ansatz functions, and efficient regularization techniques. The competitiveness of this method is shown for typical benchmark problems with up to 166 dimensions for classification in data mining, pointing out properties of sparse grids in this context. To gain insight into the adaptive refinement and to examine the scope for further improvements, the approximation of non-smooth indicator functions with adaptive sparse grids has been studied as a model problem. As an example for an improved adaptive grid refinement, we present results for an edge-detection strategy.     

An accurate gradient and Hessian reconstruction method for cell‐centered finite volume discretizations on general unstructured grids

In this paper, a novel reconstruction of the gradient and Hessian tensors on an arbitrary unstructured grid, developed for implementation in a cell‐centered finite volume framework, is presented. The reconstruction, based on the application of Gauss' theorem, provides a fully second‐order accurate estimate of the gradient, along with a first‐order estimate of the Hessian tensor. The reconstruction is implemented through the construction of coefficient matrices for the gradient components and independent components of the Hessian tensor, resulting in a linear system for the gradient and Hessian fields, which may be solved to an arbitrary precision by employing one of the many methods available for the efficient inversion of large sparse matrices. Numerical experiments are conducted to demonstrate the accuracy, robustness, and computational efficiency of the reconstruction by comparison with other common methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimate of the truncation error of finite volume discretization of the Navier–Stokes equations on colocated grids

A methodology is proposed for the calculation of the truncation error of finite volume discretizations of the incompressible Navier–Stokes equations on colocated grids. The truncation error is estimated by restricting the solution obtained on a given grid to a coarser grid and calculating the image of the discrete Navier–Stokes operator of the coarse grid on the restricted velocity and pressure field. The proposed methodology is not a new concept but its application to colocated finite volume discretizations of the incompressible Navier–Stokes equations is made possible by the introduction of a variant of the momentum interpolation technique for mass fluxes where the pressure part of the mass fluxes is not dependent on the coefficients of the linearized momentum equations. The theory presented is supported by a number of numerical experiments. The methodology is developed for two‐dimensional flows, but extension to three‐dimensional cases should not pose problems. Copyright © 2005 John Wiley & Sons, Ltd.     

An Euler system source term that develops prototype Z‐pinch implosions intended for the evaluation of shock‐hydro methods

In this paper, a phenomenological model for a magnetic drive source term for the momentum and total energy equations of the Euler system is described. This body force term is designed to produce a Z‐pinch like implosion that can be used in the development and evaluation of shock‐hydrodynamics algorithms that are intended to be used in Z‐pinch simulations. The model uses a J × B Lorentz force, motivated by a 0‐D analysis of a thin shell (or liner implosion), as a source term in the equations and allows for arbitrary current drives to be simulated. An extension that would include the multi‐physics aspects of a proposed combined radiation hydrodynamics (rad‐hydro) capability is also discussed. The specific class of prototype problems that are developed is intended to illustrate aspects of liner implosions into a near vacuum and with idealized pre‐fill plasma effects. In this work, a high‐resolution flux‐corrected‐transport method implemented on structured overlapping meshes is used to demonstrate the application of such a model to these idealized shock‐hydrodynamic studies. The presented results include an asymptotic solution based on a limiting‐case thin‐shell analytical approximation in both (x, y) and (r, z). Additionally, a set of more realistic implosion problems that include density profiles approximating plasma pre‐fill and a set of perturbed liner geometries that excite a hydro‐magnetic like Rayleigh–Taylor instability in the implosion dynamics are demonstrated. Finally, as a demonstration of including and evaluating multiphysics effects in the Euler system, a simple radiation model is included and self‐convergence results for two types of (r, z) implosions are presented. Copyright © 2008 John Wiley & Sons, Ltd.