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1.
This paper presents a detailed study on the implementation of Weighted Essentially Non‐Oscillatory (WENO) schemes on GPU. GPU implementation of up to ninth‐order accurate WENO schemes for the multi‐dimensional Euler equations of gas dynamics is presented. The implementation detail is discussed in the paper. The computational times of different schemes are obtained and the speedups are reported for different number of grid points. Furthermore, the execution times for the main kernels of the code are given and compared with each other. The numerical experiments show the speedups for the WENO schemes are very promising especially for fine grids. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

2.
Hermite weighted essentially non‐oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high‐order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193 :115). For example, classical fifth‐order weighted essentially non‐oscillatory (WENO) reconstructions are based on a five‐cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three‐cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth‐order accurate well‐balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five‐step and fourth‐order accurate method. Besides the classical SWE, the non‐homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well‐balanced treatment of the source term involved in such equations is developed and tested. Several standard one‐dimensional test cases are used to verify the high‐order accuracy, the C‐property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

3.
This paper focuses on the results of the linear stability analysis of the finite‐difference weighted essentially non‐oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order‐optimised WENO schemes of the sixth and eighth order and the bandwidth‐optimised WENO schemes of the third and fourth order are considered. Several explicit Runge–Kutta schemes including the recently published strong stability‐preserving explicit Runge–Kutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the Courant–Friedrichs–Lewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high‐order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

4.
A new third‐order WENO scheme is proposed to achieve the desired order of convergence at the critical points for scalar hyperbolic equations. A new reference smoothness indicator is introduced, which satisfies the sufficient condition on the weights for the third‐order convergence. Following the truncation error analysis, we have shown that the proposed scheme achieves the desired order accurate for smooth solutions with arbitrary number of vanishing derivatives if the parameter ε satisfies certain conditions. We have made a comparative study of the proposed scheme with the existing schemes such as WENO‐JS, WENO‐Z, and WENO‐N3 through different numerical examples. The result shows that the proposed scheme (WENO‐MN3) achieves better performance than these schemes.  相似文献   

5.
This paper proposes WCNS‐CU‐Z, a weighted compact nonlinear scheme, that incorporates adapted central difference and low‐dissipative weights together with concepts of the adaptive central‐upwind sixth‐order weighted essentially non‐oscillatory scheme (WENO‐CU) and WENO‐Z schemes. The newly developed WCNS‐CU‐Z is a high‐resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth‐order WCNS‐CU‐Z exhibits sufficient accuracy in the smooth region through use of low‐dissipative weights. The sixth‐order WCNS‐CU‐Zs are implemented with a robust linear difference formulation (R‐WCNS‐CU6‐Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that R‐WCNS‐CU6‐Z is capable of achieving a higher resolution than the seventh‐order classical robust weighted compact nonlinear scheme and can provide a crisp result in terms of discontinuity. Among the schemes tested, R‐WCNS‐CU6‐Z has been shown to be robust, and variable interpolation type R‐WCNS‐CU6‐Z (R‐WCNS‐CU6‐Z‐V) provides a stable computation by modifying the first‐order interpolation when negative density or negative pressure arises after nonlinear interpolation. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

6.
为更准确捕捉复杂流场的流动细节,通过对WENO格式的光滑因子进行改进,发展了一种新的五阶WENO格式。对三阶ENO格式进行加权可以得到五阶WENO格式,但是不同的加权处理,WENO格式在极值处保持加权基本无振荡的效果不同,本文构造了二阶精度的局部光滑因子,及不含一阶二阶导数的高阶全局光滑因子,从而实现WENO格式在极值处有五阶精度。基于改进五阶WENO格式,对一维对流方程、一维和二维可压缩无粘问题进行算例验证,并与传统WENO-JS格式和WENO-Z格式进行比较。计算结果表明,改进五阶WENO格式有较高的精度和收敛速度,有较低的数值耗散,能有效捕捉间断、激波和涡等复杂流动。  相似文献   

7.
In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

8.
We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.  相似文献   

9.
A high‐order accurate solution method for complex geometries is developed for two‐dimensional flows using the stream function–vorticity formulation. High‐order accurate spectrally optimized compact schemes along with appropriate boundary schemes are used for spatial discretization while a two‐level backward Euler implicit scheme is used for the time integration. The linear system of equations for stream function and vorticity are solved by an inner iteration while contravariant velocities constitute outer iterations. The effect of curvilinear grids on the solution accuracy is studied. The method is used to compute Cartesian and inclined driven cavity, flow in a triangular cavity and viscous flow in constricted channel. Benchmark‐like accuracy is obtained in all the problems with fewer grid points compared to reported studies. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

10.
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 ReΔx (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 Ot2) time‐integration and Ox2) 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.  相似文献   

11.
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.  相似文献   

12.
The blood flow model maintains the steady‐state solutions, in which the flux gradients are non‐zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted essentially non‐oscillatory (WENO) schemes to this model with such well‐balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well‐balanced property, and keep good resolution for smooth and discontinuous solutions. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

13.
In this paper, we propose a high‐order finite volume hybrid kinetic Weighted Essentially Non‐Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high‐order accuracy and good shock‐capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one‐dimensional and two‐dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

14.
SUMMARY

The numerical simulation of the gaseous flow originated by a gunshot both during the projectile motion within the muzzle and out of it is considered. The numerical solution of the Euler equations is obtained via a second order accurate ENO scheme. A special algorithm of cells reorganization is developed for computing the projectile motion out of the muzzle. It reveals an interesting property of the ENO schemes and Roe's approximate solution of automatically capturing discontinuities by means of the choice of the corresponding grid velocity. The algorithm developed describes sufficiently well all gasdynamics phenomena which accompany the gunshot.  相似文献   

15.
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.  相似文献   

16.
通过在单元交界面处进行高阶WENO重构,得到了一种求解双曲型守恒律方程的WENO型熵相容格式。用该格式对一维Burgers方程和Euler方程进行数值模拟,结果表明,该格式具有高精度、基本无振荡性等特点。  相似文献   

17.
By the aid of an idea of the weighted ENO schemes, some weight-type high-resolution difference schemes with different orders of accuracy are presented in this paper by using suitable weights instead of the minmod functions appearing in various TVD schemes. Numerical comparisons between the weighted schemes and the non-weighted schemes have been done for scalar equation, one-dimensional Euler equations, two-dimensional Navier-Stokes equations and parabolized Navier-Stokes equations. The project supported by the National Natural Science Foundation of China (19582007) and Partly by State Key Laboratory of Scientific/Engineering Computing.  相似文献   

18.
In this paper, the third‐order weighted essential non‐oscillatory (WENO) schemes are used to simulate the two‐dimensional shallow water equations with the source terms on unstructured meshes. The balance of the flux and the source terms makes the shallow water equations fit to non‐flat bottom questions. The simulation of a tidal bore on an estuary with trumpet shape and Qiantang river is performed; the results show that the schemes can be used to simulate the current flow accurately and catch the stronger discontinuous in water wave, such as dam break and tidal bore effectively. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

19.
This paper is concerned with the development of a high‐order upwind conservative discretization method for the simulation of flows of a Newtonian fluid in two dimensions. The fluid‐flow domain is discretized using a Cartesian grid from which non‐overlapping rectangular control volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle‐point rule. One‐dimensional integrated radial basis function schemes using the multiquadric basis function are employed to represent the variations of the field variables along the grid lines. The convection term is effectively treated using an upwind scheme with the deferred‐correction strategy. Several highly non‐linear test problems governed by the Burgers and the Navier–Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well. Copyright © 2010 John Wiley & Sons, Ltd.  相似文献   

20.
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.  相似文献   

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