High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

Improvement of multistep WENO scheme and its extension to higher orders of accuracy

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non‐oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth‐order accurate multistep methods are devised, providing weighted essentially non‐oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non‐oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock‐capturing problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐dimensional anisotropic Cartesian mesh adaptation for the compressible Euler equations

Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This paper discusses a novel two‐dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this paper, the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object‐oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant. Copyright © 2004 John Wiley & Sons, Ltd.     

Evaluation of some high‐order shock capturing schemes for direct numerical simulation of unsteady two‐dimensional free flows

The present study addresses the capability of a large set of shock‐capturing schemes to recover the basic interactions between acoustic, vorticity and entropy in a direct numerical simulation (DNS) framework. The basic dispersive and dissipative errors are first evaluated by considering the advection of a Taylor vortex in a uniform flow. Two transonic cases are also considered. The first one consists of the interaction between a temperature spot and a weak shock. This test emphasizes the capability of the schemes to recover the production of vorticity through the baroclinic process. The second one consists of the interaction of a Taylor vortex with a weak shock, corresponding to the framework of the linear theory of Ribner. The main process in play here is the production of an acoustic wave. The results obtained by using essentially non‐oscillatory (ENO), total variation diminishing (TVD), compact‐TVD and MUSCL schemes are compared with those obtained by means of a sixth‐order accurate Hermitian scheme, considered as reference. The results are as follows; the ENO schemes agree pretty well with the reference scheme. The second‐order accurate Upwind‐TVD scheme exhibits a strong numerical diffusion, while the MUSCL scheme behavior is very sensitive to the value on the parameter β in the limiter function minmod. The compact‐TVD schemes do not yield improvement over the standard TVD schemes. Copyright © 2000 John Wiley & Sons, Ltd.     

One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high‐order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p‐multigrid solution strategy has been developed, which is based on a semi‐implicit Runge–Kutta smoother for high‐order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p‐multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 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.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Performance of finite volume solutions to the shallow water equations with shock‐capturing schemes

Numerical methods have become well established as tools for solving problems in hydraulic engineering. In recent years the finite volume method (FVM) with shock capturing capabilities has come to the fore because of its suitability for modelling a variety of types of flow; subcritical and supercritical; steady and unsteady; continuous and discontinuous and its ability to handle complex topography easily. This paper is an assessment and comparison of the performance of finite volume solutions to the shallow water equations with the Riemann solvers; the Osher, HLL, HLLC, flux difference splitting (Roe) and flux vector splitting. In this paper implementation of the FVM including the Riemann solvers, slope limiters and methods used for achieving second order accuracy are described explicitly step by step. The performance of the numerical methods has been investigated by applying them to a number of examples from the literature, providing both comparison of the schemes with each other and with published results. The assessment of each method is based on five criteria; ease of implementation, accuracy, applicability, numerical stability and simulation time. Finally, results, discussion, conclusions and recommendations for further work are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

In this contribution, we investigate strategies to perform shock‐capturing computation of steady hypersonic flow fields by means of residual distribution schemes. The ultimate objective is the computation of flow solutions for which the correct upstream enthalpy value is recovered in the postshock region. To this end, the parallelism existing between the classical Bx scheme and the stabilized finite element techniques is exploited. The simple Lax‐Friedrichs dissipation term is leveraged to build two new residual distribution schemes. Upon testing on both inviscid and viscous steady problems, solutions obtained with one of the two schemes are shown to recover the correct upstream total enthalpy level in the postshock region. This last scheme provides also improved wall pressure and skin friction predictions; heat transfer predictions are, unfortunately, similar to those offered by the Bx scheme. A conjecture for explaining this behavior is exposed.     

A simple shock‐capturing technique for high‐order discontinuous Galerkin methods

This article presents a novel shock‐capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high‐order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high‐order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical investigation of a spectral‐type nodal collocation discontinuous Galerkin approximation of the Euler and Navier–Stokes equations

In this paper, we focus on the applicability of spectral‐type collocation discontinuous Galerkin methods to the steady state numerical solution of the inviscid and viscous Navier–Stokes equations on meshes consisting of curved quadrilateral elements. The solution is approximated with piecewise Lagrange polynomials based on both Legendre–Gauss and Legendre–Gauss–Lobatto interpolation nodes. For the sake of computational efficiency, the interpolation nodes can be used also as quadrature points. In this case, however, the effect of the nonlinearities in the equations and/or curved elements leads to aliasing and/or commutation errors that may result in inaccurate or unstable computations. By a thorough numerical testing on a set of well known test cases available in the literature, it is here shown that the two sets of nodes behave very differently, with a clear advantage of the Legendre–Gauss nodes, which always displayed an accurate and robust behaviour in all the test cases considered.Copyright © 2012 John Wiley & Sons, Ltd.     

Development of optimized weighted‐ENO schemes for multiscale compressible flows

The present paper deals with the development of optimized weighted–ENO schemes to improve the resolution of a class of compressible flows characterized by a wide disparity of scales, typical of compressible turbulence and/or aeroacoustic phenomena, and shock waves. The approach relies on a least square minimization of both the dispersion and dissipation error components together with the use of symmetric stencil support. Extensive numerical simulations of sound propagation, shock–sound interaction and isotropic compressible turbulence have been carried out, and the results confirm that the optimized schemes yield a resolution in wave number space greater than the non‐optimized ones. Copyright © 2003 John Wiley & Sons, Ltd.     

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

This paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

An improved unstructured WENO method for compressible multi‐fluids flow

An improved high‐order accurate WENO finite volume method based on unstructured grids for compressible multi‐fluids flow is proposed in this paper. The third‐order accuracy WENO finite volume method based on triangle cell is used to discretize the governing equations. To have higher order of accuracy, the P1 polynomial is reconstructed firstly. After that, the P2 polynomial is reconstructed from the combination of the P1. The reconstructed coefficients are calculated by analytical form of inverse matrix rather than the numerical inversion. This greatly improved the efficiency and the robustness. Four examples are presented to examine this algorithm. Numerical results show that there is no spurious oscillation of velocity and pressure across the interface and high‐order accurate result can be achieved. Copyright © 2016 John Wiley & Sons, Ltd.     

An adaptive least‐squares method for the compressible Euler equations

An adaptive least‐squares finite element method is used to solve the compressible Euler equations in two dimensions. Since the method is naturally diffusive, no explicit artificial viscosity is added to the formulation. The inherent artificial viscosity, however, is usually large and hence does not allow sharp resolution of discontinuities unless extremely fine grids are used. To remedy this, while retaining the advantages of the least‐squares method, a moving‐node grid adaptation technique is used. The outstanding feature of the adaptive method is its sensitivity to directional features like shock waves, leading to the automatic construction of adapted grids where the element edge(s) are strongly aligned with such flow phenomena. Using well‐known transonic and supersonic test cases, it has been demonstrated that by coupling the least‐squares method with a robust adaptive method shocks can be captured with high resolution despite using relatively coarse grids. Copyright © 1999 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

Performance of very‐high‐order upwind schemes for DNS of compressible wall‐turbulence

The purpose of the present paper is to evaluate very‐high‐order upwind schemes for the direct numerical simulation (DNS ) of compressible wall‐turbulence. We study upwind‐biased (UW ) and weighted essentially nonoscillatory (WENO ) schemes of increasingly higher order‐of‐accuracy (J. Comp. Phys. 2000; 160 :405–452), extended up to WENO 17 (AIAA Paper 2009‐1612, 2009). Analysis of the advection–diffusion equation, both as Δx→0 (consistency), and for fixed finite cell‐Reynolds‐number Re_Δx (grid‐resolution), indicates that the very‐high‐order upwind schemes have satisfactory resolution in terms of points‐per‐wavelength (PPW ). Computational results for compressible channel flow (Re∈[180, 230]; M?_CL ∈[0.35, 1.5]) are examined to assess the influence of the spatial order of accuracy and the computational grid‐resolution on predicted turbulence statistics, by comparison with existing compressible and incompressible DNS databases. Despite the use of baseline O(Δt²) time‐integration and O(Δx²) discretization of the viscous terms, comparative studies of various orders‐of‐accuracy for the convective terms demonstrate that very‐high‐order upwind schemes can reproduce all the DNS details obtained by pseudospectral schemes, on computational grids of only slightly higher density. Copyright © 2009 John Wiley & Sons, Ltd.     

An eigenvector‐based linear reconstruction scheme for the shallow‐water equations on two‐dimensional unstructured meshes

This paper presents a new approach to MUSCL reconstruction for solving the shallow‐water equations on two‐dimensional unstructured meshes. The approach takes advantage of the particular structure of the shallow‐water equations. Indeed, their hyperbolic nature allows the flow variables to be expressed as a linear combination of the eigenvectors of the system. The particularity of the shallow‐water equations is that the coefficients of this combination only depend upon the water depth. Reconstructing only the water depth with second‐order accuracy and using only a first‐order reconstruction for the flow velocity proves to be as accurate as the classical MUSCL approach. The method also appears to be more robust in cases with very strong depth gradients such as the propagation of a wave on a dry bed. Since only one reconstruction is needed (against three reconstructions in the MUSCL approach) the EVR method is shown to be 1.4–5 times as fast as the classical MUSCL scheme, depending on the computational application. Copyright © 2006 John Wiley & Sons, Ltd.