WENO Method Research Articles

High-order ALE gas-kinetic scheme with WENO reconstruction

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities.

An improved WLS-ENO method for solving hyperbolic conservation laws

Liu and Jiao proposed a WLS-ENO method (Weighted-Least-Squares based Essentially Non-Oscillatory). Comparing to the WENO method (Weighted ENO), the WLS-ENO method is more flexible for solving the hyperbolic conservation laws. Notice that, a WLS problem is generated for each WLS-ENO procedure, although the QR factorization with column pivoting is employed, the computational cost of solving WLS problem is still heavy. So in this paper, we will propose an improved WLS-ENO method for improving the efficiency. The basic idea of new method is that, let the reconstructed polynomial approximate the cell averages of some smooth cells in conservation fashion, and approximate other cell averages in WLS fashion, so that the resulted WLS problem only has one unknown. Because the resulted WLS problem can be solved by using the normal equation directly, the whole computational cost is small, thus the new method has better efficiency.

Lattice Boltzmann models based on the vielbein formalism for the simulation of flows in curvilinear geometries

In this paper, we consider the Boltzmann equation with respect to orthonormal vielbein fields in conservative form. This formalism allows the use of arbitrary coordinate systems to describe the space geometry, as well as of an adapted coordinate system in the momentum space, which is linked to the physical space through the use of vielbeins. Taking advantage of the conservative form, we derive the macroscopic equations in a covariant tensor notation, and show that the hydrodynamic limit can be obtained via the Chapman-Enskog expansion in the Bhatnaghar-Gross-Krook approximation for the collision term. We highlight that in this formalism, the component of the momentum which is perpendicular to some curved boundary can be isolated as a separate momentum coordinate, for which the half-range Gauss-Hermite quadrature can be applied. We illustrate the capabilities of this formalism by considering two applications. The first one is the circular Couette flow between rotating coaxial cylinders, for which benchmarking data are available for all degrees of rarefaction, from the hydrodynamic to the ballistic regime. The second application concerns the flow in a gradually expanding channel. We employ finite-difference lattice Boltzmann models based on half-range Gauss-Hermite quadratures for the implementation of diffuse reflection, together with the fifth-order weighted essentially nonoscillatory and third-order total variation diminishing Runge-Kutta numerical methods for the advection and time stepping, respectively.

A moving mesh WENO method based on exponential polynomials for one-dimensional conservation laws

In the article [Yang et al. (2012) [37]], the authors have developed a high order moving mesh WENO method for one-dimensional (1D) hyperbolic conservation laws, which is shown to be effective in resolving shocks and other complex solution structures. In this paper, in the light of the similar moving mesh technique, we develop a novel WENO scheme with non-polynomial bases, in particular, the exponential bases to further improve the performance of WENO schemes for solving 1D conservation laws. Furthermore, we modify the original moving mesh technique by developing a new monitor function as well as a different mesh smoothing strategy. A collection of numerical examples is presented to demonstrate high order accuracy and robustness of the method in capturing smooth and non-smooth solutions including the strong δ shock arising from the weakly hyperbolic pressureless Euler equations.

Unsteady Discontinuous Galerkin Method of a High Order of Accuracy for Modeling Turbulent Flows

This paper presents a brief description of the Zhukovsky Central Aerohydrodynamic Institute (TsAGI) code based on the Galerkin method of a high order of accuracy with discontinuous basic functions. The functions is reconstructed for conservative variables. Gradients of the variables are determined using the Bassi–Rebay 2 method. For integrating, the Gaussian quadrature rules are used. The coordinates are transformed by serendipian elements. In computing by schemes of an order higher than the second order, the curvature of grid lines is taken into account. A comparison with finite volume methods is performed, including the WENO method with constant weights and a single quadrature point on a cell face. The classical tests are used, namely, a subsonic flow around a circular cylinder in an ideal gas, the diagonal convection of a two-dimensional isentropic vortex, and the decay of the Taylor–Green vortex.

IB-WENO method for incompressible flow with elastic boundaries

Abstract We present a new approach to compute the interaction between an elastic boundary and an incompressible viscous fluid, by integrating the immersed boundary method and a finite difference WENO scheme. The hybrid algorithm maintains simple and efficient treatment of the moving interface based on the immersed boundary formulation, and achieves high-order accuracy, with the WENO method and the Hodge decomposition technique employed, in the majority of the fluid domain that does not contain the interface. We demonstrate the performance of the proposed algorithm and its various numerical components through a number of computational examples.

Mapped CWENO scheme for hyperbolic conservation laws

In this paper, we analysis the Central Weighted Essentially Non-Oscillatory (CWENO) scheme for hyperbolic conservation laws (Qiu and Shu, 2002). The CWENO procedure contains three major steps: the reconstruction of u ̄ i + 1 ∕ 2 n from u ̄ i n , the reconstruction of u i n from u ̄ i n , and the approximation of ∫ t n t n + 1 f ( u ( x i , t ) ) d t using the Natural Continuous Extension (NCE) of Runge–Kutta method. We found that both reconstructions lost the designed optimal order of accuracy near critical points. This loss of accuracy can be recovered by using a proper mapping function. Furthermore, numerical tests demonstrate that, in practice, it is enough for the mapping technique to be implemented only in the first reconstruction. This new mapped central WENO method is not only more accurate than the one in Qiu and Shu (2002), but also more efficient.

The Approximated Semi-Lagrangian WENO Methods Based on Flux Vector Splitting for Hyperbolic Conservation Laws

The paper is devised to combine the approximated semi-Lagrange weighted essentially non-oscillatory scheme and flux vector splitting. The approximated finite volume semi-Lagrange that is weighted essentially non-oscillatory scheme with Roe flux had been proposed. The methods using Roe speed to construct the flux probably generates entropy-violating solutions. More seriously, the methods maybe perform numerical instability in two-dimensional cases. A robust and simply remedy is to use a global flux splitting to substitute Roe flux. The combination is tested by several numerical examples. In addition, the comparisons of computing time and resolution between the classical weighted essentially non-oscillatory scheme (WENOJS-LF) and the semi-Lagrange weighted essentially non-oscillatory scheme (WENOEL-LF) which is presented (both combining with the flux vector splitting).

Conservative and Easily Implemented Finite Volume Semi-Lagrangian WENO Methods for 1D and 2D Hyperbolic Conservation Laws

The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.

Embedded WENO: A design strategy to improve existing WENO schemes

Embedded WENO methods utilise all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu [1] and on the WENO-Z scheme of Borges et al. [2] are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.

NUMERICAL SIMULATION OF THE SVS 13 MICROJET AND BOW SHOCK BUBBLE

ABSTRACT Numerical simulations of the SVS 13 microjet and bow shock bubble are performed using the WENO method that reproduces the main features and dynamics of data from the Keck Telescope/OSIRIS velocity-resolved integral field spectrograph: an expanding, cooler bow shock bubble, with the bubble center moving at approximately 50 km s−1 with a radial expansion velocity of 11 km s−1, surrounding the fast, hotter jet, which is propagating at 156 km s−1. Contact and bow shock waves are visible in the simulations both from the initial short jet pulse that creates the nearly spherical bow shock bubble and from the fast microjet, while a terminal Mach disk shock is visible near the tip of the continuous microjet, which reduces the velocity of the jet gas down to the flow velocity of the contact discontinuity at the leading edge of the jet. At 21.1 years after the launch of the initial bubble pulse, the jet has caught up with and penetrated almost all the way across the bow shock bubble of the slower initial pulse. At times later than about 22 years, the jet has penetrated through the bubble and thereafter begins to subsume its spherical form. Emission maps from the simulations of the jet—traced by the emission of the shock-excited 1.644 μm [Fe ii] line—and the bow shock bubble—traced in the lower excitation 2.122 μm H2 1–0 S(1) line—projected onto the plane of the sky are presented, and are in good agreement with the Keck observations.

Solving 3D relativistic hydrodynamical problems with weighted essentially nonoscillatory discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high order convergence for smooth problems and for the simulation of discontinuities and shocks. In this work, we investigate WENO-DG algorithms in the context of numerical general relativity, in particular for general relativistic hydrodynamics. We implement the standard WENO method at different orders, a compact (simple) WENO scheme, as well as an alternative subcell evolution algorithm. To evaluate the performance of the different numerical schemes, we study non-relativistic, special relativistic, and general relativistic testbeds. We present the first three-dimensional simulations of general relativistic hydrodynamics, albeit for a fixed spacetime background, within the framework of WENO-DG methods. The most important testbed is a single TOV-star in three dimensions, showing that long term stable simulations of single isolated neutron stars can be obtained with WENO-DG methods.

High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study

AbstractFixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied in solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.

High-order central Hermite WENO schemes: Dimension-by-dimension moment-based reconstructions

In this paper, a class of high-order central finite volume schemes is proposed for solving one- and two-dimensional hyperbolic conservation laws. Formulated on staggered meshes, the methods involve Hermite WENO (HWENO) spatial reconstructions, and Lax–Wendroff type discretizations or the natural continuous extension of Runge–Kutta methods in time. Differently from the central Hermite WENO methods we developed previously in Tao et al. (2015) [34] , the spatial reconstructions, a core ingredient of the methods, are based on the zeroth-order and the first-order moments of the solution, and are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. This leads to much simpler implementation of the methods in higher dimension and better cost efficiency. Meanwhile, the proposed methods have the attractive features of the general central Hermite WENO methods such as being compact in reconstruction and requiring neither flux splitting nor numerical fluxes, while being accurate and essentially non-oscillatory. A collection of one- and two-dimensional numerical examples is presented to demonstrate high resolution and robustness of the methods in capturing smooth and non-smooth solutions.

An improved unstructured WENO method for compressible multi‐fluids flow

SummaryAn 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.

A computational approach to flame hole dynamics using an embedded manifold approach

We present a new numerical algorithm for tracking the evolution of flame holes in diffusion flames. The key element is the solution of an evolution equation for a flame state field defined on a complex moving surface. The surface itself can evolve in time and is defined implicitly as a level set of an associated Cartesian scalar field. The surface coordinates, or parameterization, do not need to be determined explicitly. Instead, the numerical method employs an embedding technique where the evolution equation is extended to the Cartesian space. In our application, the flame state field represents the chemical activity state of a diffusion flame; i.e. quenched and burning regions of the flame surface. We present a formulation that describes the formation, propagation, and growth of flame holes using edge-flame modeling in laminar and turbulent diffusion flames. The evolution equation is solved using a high-order finite-volume WENO method and a new extension algorithm defined in terms of propagation PDEs. The complete algorithm is demonstrated by tracking the dynamics of flame holes in a turbulent reacting shear layer and its applicability is also demonstrated in a turbulent reacting lifted jet simulation.

Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

We present a WENO finite volume method for the approximation of hyperbolic conservation laws on adaptively refined Cartesian grids.On each single patch of the AMR grid, we use a modified dimension-by-dimension WENO method, which was recently developed by Buchmüller and Helzel (2014) [1]. This method retains the full spatial order of accuracy of the underlying one-dimensional WENO reconstruction for nonlinear multidimensional problems, and requires only one flux computation per interface. It is embedded into block-structured AMR through conservative interpolation functions and a numerical flux fix that transfers data between different levels of grid refinement.Numerical tests illustrate the accuracy of the new adaptive WENO finite volume method. Compared to the classical dimension-by-dimension approach, the new method is much more accurate while it is only slightly more expensive. Furthermore, we also show results of an accuracy study for an adaptive WENO method which uses multidimensional reconstruction of the conserved quantities and a high-order quadrature formula to compute the fluxes. While the accuracy of such a method is comparable with our new approach, it is about three times more expensive than the latter.

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

In this paper we present a new and efficient quadrature-free formulation for the family of cell-centered high order accurate direct arbitrary-Lagrangian---Eulerian one-step ADER-WENO finite volume schemes on unstructured triangular and tetrahedral meshes that has been developed by the authors in a recent series of papers (Boscheri et al. in J Comput Phys 267:112---138, 2014; Boscheri and Dumbser in Commun Comput Phys 14:1174---1206, 2013; Boscheri and Dumbser in J Comput Phys 275:484---523, 2014; Dumbser and Boscheri in Comput Fluids 86:405---432, 2013). High order of accuracy in time is obtained by using a local space---time Galerkin predictor on moving curved meshes, while a high order accurate nonlinear WENO method is adopted to produce high order essentially non-oscillatory reconstruction polynomials in space. The mesh is moved at each time step according to the solution of a node solver algorithm that assigns a unique velocity vector to each node of the mesh. A rezoning procedure can also be applied when mesh distortions and deformations become too severe. The space---time mesh is then constructed by straight edges connecting the vertex positions at the old time level $$t^n$$tn with the new ones at the next time level $$t^{n+1}$$tn+1, yielding closed space---time control volumes, on the boundary of which the numerical flux must be integrated. This is done here with a new and efficient quadrature-free approach: the space---time boundaries are split into simplex sub-elements, i.e. either triangles in 2D or tetrahedra in 3D. This leads to space---time normal vectors as well as Jacobian matrices that are constant within each sub-element. Within the space---time Galerkin predictor stage that solves the Cauchy problem inside each element in the small, the discrete solution and the flux tensor are approximated using a nodal space---time basis. Since these space---time basis functions are defined on a reference element and do not change, their integrals over the simplex sub-surfaces of the space---time reference control volume can be integrated once and for all analytically during a preprocessing step. The resulting integrals are then used together with the space---time degrees of freedom of the predictor in order to compute the numerical flux that is needed in the finite volume scheme. We apply the high order algorithm presented in this paper to the equations of hydrodynamics obtaining convergence rates up to fourth order of accuracy in space and time. A set of classical Lagrangian test problems has been solved and the results have been compared with the ones given by the original formulation of the algorithm (Boscheri and Dumbser 2013, 2014). The efficiency has been monitored and measured for each test case and the new quadrature-free schemes were up to 3.7 times faster than the ones based on Gaussian quadrature.

Higher-order finite volume method with semi-Lagrangian scheme for one-dimensional conservation laws

In this paper, a high-order, semi-Lagrangian finite volume (SL-FV) method based on the WENO approach is proposed in order to manage one-dimensional conservation laws. The proposed method successfully integrates WENO reconstructions and the semi-Lagrangian method. More specifically, the Taylor expansion of time is used to approximate the time integration, deployed to boost temporal accuracy. Next, characteristic curves are applied to replace the time level by points in the semi-Lagrangian method. The value of these points can then be reconstructed by WENO schemes to increase their accuracy in space. Both high-order accuracies in space and time, respectively, are achieved. Moreover, computational experiments allow for a weaker CFL condition, provided in detail to validate the performance of the proposed SL-FV-based WENO method.

Progress in high-resolution numerical simulation of explosion mechanics

nitride semiconductor material, microwave power device, power electronic device High-resolution numerical simulation of explosion problems is of significant theoretical value owing to the numerous potential applications in both national defense and civilian projects, such as ammunition design, damage assessment, target protection, and industrial gas explosion prevention. In this paper, we review the research history of explosion mechanics and present the highest-order numerical simulation in decades. Building upon the authors research work in recent years, we specifically discuss the development of high-order positivity-preserving schemes, high-order boundary treatment, an adaptive mesh method, and multi-material interface treatment. We extended the WENO and RKDG methods and constructed conservative positivity-preserving high-order schemes. The problems of negative density and pressure were solved through the simulation of complex detonation propagation. The detonation wave front structure and flow characteristics were captured clearly. In this paper, a high-order inverse Lax-Wendroff method on a Cartesian mesh is presented to treat the complex boundary. The main principle is that the first-order normal derivative can be obtained by the inverse Lax-Wendroff method, while all the higher-order normal derivatives can simply be obtained by fifth-order WENO-type extrapolation. The efficient implementation in the detonation process further indicates that the inverse Lax-Wendroff method is stable and robust when applied to a complex-geometry boundary. An a posteriori error estimate, which is often used as a refinement criterion, is applied to solve the Euler equations. To achieve mesh refinement, WENO and Hermite interpolations are used to prolong the solutions from coarse to fine grid in space and time, respectively. For mesh merging, the direct point value replacement method is employed to update the solutions of coarse grids. For this study, a local level-set tracking method was developed to treat the multi- material interface, and the velocity field in the computational domain was modified in order to effectively avoid errors caused by velocity field discontinuity when the interface position was solved. The method presented in this paper combines the GFM (Ghost Fluid Method) with the RGFM (Real Ghost Fluid Method) to deal with the material interface, thereby avoiding non-physical solutions when the GFM is used to deal with a strong discontinuity, and the CPU running time is increased when RGFM is employed to deal with a weak discontinuity in solving the Riemann problem repeatedly. Finally, the future of high-order numerical simulation of explosion mechanics is discussed.