Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

求解双曲守恒律方程的WENO型熵相容格式

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。

     

Composite high resolution localized relaxation scheme based on upwinding for hyperbolic conservation laws

In this work we present an upwind‐based high resolution scheme using flux limiters. Based on the direction of flow we choose the smoothness parameter in such a way that it leads to a truly upwind scheme without losing total variation diminishing (TVD) property for hyperbolic linear systems where characteristic values can be of either sign. Here we present and justify the choice of smoothness parameters. The numerical flux function of a high resolution scheme is constructed using wave speed splitting so that it results into a scheme that truly respects the physical hyperbolicity property. Bounds are given for limiter functions to satisfy TVD property. The proposed scheme is extended for non‐linear problems by using the framework of relaxation system that converts a non‐linear conservation law into a system of linear convection equations with a non‐linear source term. The characteristic speed of relaxation system is chosen locally on three point stencil of grid. This obtained relaxation system is solved using composite scheme technique, i.e. using a combination of proposed scheme with the conservative non‐standard finite difference scheme. Presented numerical results show higher resolution near discontinuity without introducing spurious oscillations. Copyright © 2008 John Wiley & Sons, Ltd.     

An improved third-order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L_∞ and L₂ norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes.     

Non-oscillatory central-upwind scheme for hyperbolic conservation laws

We introduce a new fourth order, semi-discrete, central-upwind scheme for solving systems of hyperbolic conservation laws. The scheme is a combination of a fourth order non-oscillatory reconstruction, a semi-discrete central-upwind numerical flux and the third order TVD Runge-Kutta method. Numerical results suggest that the new scheme achieves a uniformly high order accuracy for smooth solutions and produces non-oscillatory profiles for discontinuities. This is especially so for long time evolution problems. The scheme combines the simplicity of the central schemes and accuracy of the upwind schemes. The advantages of the new scheme will be fully realized when solving various examples.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

A new high-accuracy scheme for compressible turbulent flows

Shock-capturing and broad-bandwidth scale resolutions are two main challenges of compressible turbulent flow simulation. To meet the rigorous requests, a novel fifth-order hybrid scheme based on a uniform hybrid framework is designed. With the help of a continuous weight operator, the new scheme combines an upwind compact scheme for smooth regions and a compact-reconstruction weighted essentially non-oscillatory scheme for discontinuous regions. Numerical analyses and canonical numerical tests confirm that the new scheme has high accuracy, spectral-like resolution property and shock-capturing capability. Besides, the new scheme shows high computational efficiency compared to the related shock-capturing schemes and hybrid ones.     

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

This paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.     

A semi‐discrete central scheme for scalar hyperbolic conservation laws with heterogeneous storage coefficient and its application to porous media flow

In this paper, we develop a new Godunov‐type semi‐discrete central scheme for a scalar conservation law on the basis of a generalization of the Kurganov and Tadmor scheme, which allows for spatial variability of the storage coefficient (e.g. porosity in multiphase flow in porous media) approximated by piecewise constant interpolation. We construct a generalized numerical flux at element edges on the basis of a nonstaggered inhomogeneous dual mesh, which reproduces the one postulated by Kurganov and Tadmor under the assumption of homogeneous storage coefficient. Numerical simulations of two‐phase flow in strongly heterogeneous porous media illustrate the performance of the proposed scheme and highlight the important rule of the permeability–porosity correlation on finger growth and breakthrough curves. Copyright © 2013 John Wiley & Sons, Ltd.     

A C2‐continuous high‐resolution upwind convection scheme

A bounded upwinding scheme for numerical solution of hyperbolic conservation laws and Navier–Stokes equations is presented. The scheme is based on convection boundedness criterion and total variation diminishing stability criteria and developed by employing continuously differentiable functions. The accuracy of the scheme is verified by assessing the error and observed convergence rate on 1‐D benchmark test cases. A comparative study between the new scheme and conventional total variation diminishing/convection boundedness criterion‐based upwind schemes to solve standard nonlinear hyperbolic conservation laws is also accomplished. The scheme is then examined in the simulation of Newtonian and non‐Newtonian fluid flows of increasing complexity; a satisfactory agreement has been observed in terms of the overall behavior. Finally, the scheme is used to study the hydrodynamics of a gas‐solid flow in a bubbling fluidized bed. Copyright © 2013 John Wiley & Sons, Ltd.     

求解双曲守恒律的修正模板近似的五阶WENO格式

针对经典的五阶加权本质无振荡(WENO)格式在间断附近耗散过大以及临界点不能保精度的问题,本文提出了一种新的修正模板近似方法。改进了经典五阶WENO-JS格式中各候选子模板上数值通量的二阶多项式逼近,通过加入三次修正项使模板逼近达到四阶精度,并且通过引入可调函数φ使得新的格式具有ENO性质,理论分析新的格式具有保精度特性,通过一系列数值算例说明了新格式的高效性。     

A time‐implicit high‐order compact differencing and filtering scheme for large‐eddy simulation

This work investigates a high‐order numerical method which is suitable for performing large‐eddy simulations, particularly those containing wall‐bounded regions which are considered on stretched curvilinear meshes. Spatial derivatives are represented by a sixth‐order compact approximation that is used in conjunction with a tenth‐order non‐dispersive filter. The scheme employs a time‐implicit approximately factored finite‐difference algorithm, and applies Newton‐like subiterations to achieve second‐order temporal and sixth‐order spatial accuracy. Both the Smagorinsky and dynamic subgrid‐scale stress models are incorporated in the computations, and are used for comparison along with simulations where no model is employed. Details of the method are summarized, and a series of classic validating computations are performed. These include the decay of compressible isotropic turbulence, turbulent channel flow, and the subsonic flow past a circular cylinder. For each of these cases, it was found that the method was robust and provided an accurate means of describing the flowfield, based upon comparisons with previous existing numerical results and experimental data. Published in 2003 by John Wiley & Sons, Ltd.     

A new smoothness indicator for third‐order WENO scheme

We present a new reference smoothness indicator for third‐order weighted essentially non‐oscillatory scheme to recover its design‐order convergence at critical points. This reference smoothness indicator, which involves both the candidate and global smoothness indicators in the weighted essentially non‐oscillatory framework, is devised according to a sufficient condition on the weights for third‐order convergence. The recovery of design‐order is verified by standard tests. Meanwhile, numerical results demonstrate that the present reference smoothness indicator produces sharper representation of the discontinuity owing to the combined effects of larger weight assignment to the discontinuous stencils and convergence rate recovery. Copyright © 2015 John Wiley & Sons, Ltd.     

基于特征分裂改进的WENO格式与Roe格式对比研究

双曲性守恒方程组采用高阶、高分辨率的WENO格式时有两类分裂方法,即逐点分裂和特征分裂。本文基于后者,对特征分裂重构时强间断和接触间断位置出现的振荡情况进行研究,对重构变量加以改进,发现改进后的WENO格式克服了间断处的振荡,然后以LU-SGS为子迭代的双时间步法求解Euler方程,选用一维Sod、二维前台阶和双马赫反射算例,并与Roe格式计算结果进行对比,发现WENO格式分辨率更高,耗散更小。     

A high‐resolution scheme for compressible multicomponent flows with shock waves

A simple methodology for a high‐resolution scheme to be applied to compressible multicomponent flows with shock waves is investigated. The method is intended for use with direct numerical simulation or large eddy simulation of compressible multicomponent flows. The method dynamically adds non‐linear artificial diffusivity locally in space to capture different types of discontinuities such as a shock wave, contact surface or material interface while a high‐order compact differencing scheme resolves a broad range of scales in flows. The method is successfully applied to several one‐dimensional and two‐dimensional compressible multicomponent flow problems with shock waves. The results are in good agreement with experiments and earlier computations qualitatively and quantitatively. The method captures unsteady shock and material discontinuities without significant spurious oscillations if initial start‐up errors are properly avoided. Comparisons between the present numerical scheme and high‐order weighted essentially non‐oscillatory (WENO) schemes illustrate the advantage of the present method for resolving a broad range of scales of turbulence while capturing shock waves and material interfaces. Also the present method is expected to require less computational cost than popular high‐order upwind‐biased schemes such as WENO schemes. The mass conservation for each species is satisfied due to the strong conservation form of governing equations employed in the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive mesh refinement for high‐resolution finite element schemes

New a posteriori error indicators based on edgewise slope‐limiting are presented. The L₂‐norm is employed to measure the error of the solution gradient in both global and element sense. A second‐order Newton–Cotes formula is utilized in order to decompose the local gradient error from a ??₁ finite element solution into a sum of edge contributions. The slope values at edge midpoints are interpolated from the two adjacent vertices. Traditional techniques to recover (superconvergent) nodal gradient values from consistent finite element slopes are reviewed. The deficiencies of standard smoothing procedures—L₂‐projection and the Zienkiewicz–Zhu patch recovery—as applied to nonsmooth solutions are illustrated for simple academic configurations. The recovered gradient values are corrected by applying a slope limiter edge‐by‐edge so as to satisfy geometric constraints. The direct computation of slopes at edge midpoints by means of limited averaging of adjacent gradient values is proposed as an inexpensive alternative. Numerical tests for various solution profiles in one and two space dimensions are presented to demonstrate the potential of this postprocessing procedure as an error indicator. Finally, it is used to perform adaptive mesh refinement for compressible inviscid flow simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.