High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 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 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.     

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

We discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi‐steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well‐balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed–load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump‐shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi‐steady approach only when the bed–load rapidly interacts with the hydrodynamics. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Finite volume scheme for the solution of fluid flow problems on unstructured non‐staggered grids

A recently developed non‐staggered methodology which uses the principle of applying fourth‐order dissipation to the governing pressure‐correction equation is developed so it can be applied to unstructured grids. A finite volume methodology is used for discretization. The fourth‐order dissipation term is found using second‐order gradient operators. This makes it straightforward to incorporate the dissipation term on unstructured grids. The new methodology is compared with solutions from a standard finite volume second‐order flow solver and is also tested for a standard laminar driven‐lid flow problem with grids systems that do not have a uniform structure. Finally, we demonstrate how the new methodology can be used to predict flow over a wavy boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids

In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.     

Characteristic-based finite volume scheme for 1D Euler equations

In this paper, a high-order finite-volume scheme is presented for the one- dimensional scalar and inviscid Euler conservation laws. The Simpson＇s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson＇s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory （CWENO） reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.     

Hermite WENO-based limiters for high order discontinuous Galerkin method on unstructured grids

A novel class of weighted essentially nonoscillatory (WENO) schemes based on Hermite polynomials,termed as HWENO schemes,is developed and applied as limiters for high order discontinuous Galerkin (DG) method on triangular grids.The developed HWENO methodology utilizes high-order derivative information to keep WENO reconstruction stencils in the von Neumann neighborhood.A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils,making higher-order scheme stable and simplifying the reconstruction process at the same time.The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement.Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy,the designed HWENO limiters can simultaneously obtain uniform high order accuracy and sharp,essentially non-oscillatory shock transition.     

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Numerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite volume scheme for solving anisotropic diffusion on ALE‐AMR grids

In the context of High Energy Density Physics and more precisely in the field of laser plasma interaction, Lagrangian schemes are commonly used. The lack of robustness due to strong grid deformations requires the regularization of the mesh through the use of Arbitrary Lagrangian Eulerian methods. Theses methods usually add some diffusion and a loss of precision is observed. We propose to use Adaptive Mesh Refinement (AMR) techniques to reduce this loss of accuracy. This work focuses on the resolution of the anisotropic diffusion operator on Arbitrary Lagrangian Eulerian‐AMR grids. In this paper, we describe a second‐order accurate cell‐centered finite volume method for solving anisotropic diffusion on AMR type grids. The scheme described here is based on local flux approximation which can be derived through the use of a finite difference approximation, leading to the CCLADNS scheme. We present here the 2D and 3D extension of the CCLADNS scheme to AMR meshes. Copyright © 2017 John Wiley & Sons, Ltd.     

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

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.     

A deconvolution‐based fourth‐order finite volume method for incompressible flows on non‐uniform grids

This paper is concerned with the development of a new high‐order finite volume method for the numerical simulation of highly convective unsteady incompressible flows on non‐uniform grids. Specifically, both a high‐order fluxes integration and the implicit deconvolution of the volume‐averaged field are considered. This way, the numerical solution effectively stands for a fourth‐order approximation of the point‐wise one. Moreover, the procedure is developed in the framework of a projection method for the pressure–velocity decoupling, while originally deriving proper high‐order intermediate boundary conditions. The entire numerical procedure is discussed in detail, giving particular attention to the consistent discretization of the deconvolution operation. The present method is also cast in the framework of approximate deconvolution modelling for large‐eddy simulation. The overall high accuracy of the method, both in time and space, is demonstrated. Finally, as a model of real flow computation, a two‐dimensional time‐evolving mixing layer is simulated, with and without sub‐grid scales modelling. Copyright © 2003 John Wiley & Sons, Ltd.     

Well‐balanced discontinuous Galerkin method and finite volume WENO scheme based on hydrostatic reconstruction for blood flow model in arteries

The blood flow model in arteries admits the steady state solutions, for which the flux gradient is nonzero, and is exactly balanced by the source term. In this paper, by means of hydrostatic reconstruction, we construct a high order discontinuous Galerkin method, which exactly preserves the dead‐man steady state, which is characterized by a discharge equal to zero (analogue to hydrostatic equilibrium). Moreover, the method maintains genuine high order of accuracy. Subsequently, we apply the key idea to finite volume weighted essentially non‐oscillatory schemes and obtain a well‐balanced finite volume weighted essentially non‐oscillatory scheme. Extensive numerical experiments are performed to verify the well‐balanced property, high order accuracy, as well as good resolution for smooth and discontinuous solutions.     

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.     

Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

In shallow water flow and transport modeling, the monotonic upstream‐centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second‐order accuracy. The most important step in MUSCL‐type schemes is MUSCL reconstruction, which calculate‐extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2‐point stencil that uses the cell centers of the neighbor cell and the so‐called far‐neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.