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.     

Improved total variation diminishing schemes for advection simulation on arbitrary grids

The requirements for flux limiter functions preserving total variation diminishing (TVD) are derived based on a 1D nonuniform grid, and a new TVD region is determined to fit arbitrary 1D grids. Some second‐order TVD schemes called improved TVD schemes are developed, such as modified Van Leer scheme, modified Van Albada scheme, and modified SUPERBEE scheme. Then, they are extended to 2D grids. Because the element sizes and face positions are taken into account, good behaviors are observed in the implementations in both 1D and 2D cases for pure advection simulation. That is, good conservation, better monotonicity, and higher accuracy are maintained by the improved TVD schemes compared with the present ones deduced from uniform grids, and they keep superiorities even when implemented on poor grids. Among all the improved TVD schemes, the modified SUPERBEE is only recommended for poor 1D grids, but the modified Van Leer scheme can suit both poor 1D and 2D grids. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient construction of high‐resolution TVD conservative schemes for equations with source terms: application to shallow water flows

High‐resolution total variation diminishing (TVD) schemes are widely used for the numerical approximation of hyperbolic conservation laws. Their extension to equations with source terms involving spatial derivatives is not obvious. In this work, efficient ways of constructing conservative schemes from the conservative, non‐conservative or characteristic form of the equations are described in detail. An upwind, as opposed to a pointwise, treatment of the source terms is adopted here, and a new technique is proposed in which source terms are included in the flux limiter functions to get a complete second‐order compact scheme. A new correction to fix the entropy problem is also presented and a robust treatment of the boundary conditions according to the discretization used is stated. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical study of wave propagation in compressible two‐phase flow

We propose a new model and a solution method for two‐phase two‐fluid compressible flows. The model involves six equations obtained from conservation principles applied to a one‐dimensional flow of gas and liquid mixture completed by additional closure governing equations. The model is valid for pure fluids as well as for fluid mixtures. The system of partial differential equations with source terms is hyperbolic and has conservative form. Hyperbolicity is obtained using the principles of extended thermodynamics. Features of the model include the existence of real eigenvalues and a complete set of independent eigenvectors. Its numerical solution poses several difficulties. The model possesses a large number of acoustic and convective waves and it is not easy to upwind all of these accurately and simply. In this paper we use relatively modern shock‐capturing methods of a centred‐type such as the total variation diminishing (TVD) slope limiter centre (SLIC) scheme which solve these problems in a simple way and with good accuracy. Several numerical test problems are displayed in order to highlight the efficiency of the study we propose. The scheme provides reliable results, is able to compute strong shock waves and deals with complex equations of state. Copyright © 2006 John Wiley & Sons, Ltd.     

A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 John Wiley & Sons, Ltd.     

A variable relaxation scheme for multiphase,multicomponent flow

We propose a variable relaxation scheme for the simulation of 1D, two-phase, multicomponent flow in porous media. For these strongly nonlinear systems, traditional high order upwind schemes are impractical: Riemann solutions are not directly available when the phase behavior is complex, and the systems are weakly hyperbolic at isolated points. Relaxation schemes avoid the dependency on the eigenstructure and nonlinear Riemann solvers by approximating the original system with a strongly hyperbolic linear system. We exploit the known information about the eigenvalues to construct first order and second order variable relaxation schemes with much reduced numerical diffusion as compared to the standard relaxation formulations. The proposed second order variable relaxation scheme is competitive in accuracy and efficiency with a third order component-wise ENO reconstruction, and performs at least as well as second order component-wise TVD schemes.     

Positivity‐preserving,flux‐limited finite‐difference and finite‐element methods for reactive transport

A new class of positivity‐preserving, flux‐limited finite‐difference and Petrov–Galerkin (PG) finite‐element methods are devised for reactive transport problems.The methods are similar to classical TVD flux‐limited schemes with the main difference being that the flux‐limiter constraint is designed to preserve positivity for problems involving diffusion and reaction. In the finite‐element formulation, we also consider the effect of numerical quadrature in the lumped and consistent mass matrix forms on the positivity‐preserving property. Analysis of the latter scheme shows that positivity‐preserving solutions of the resulting difference equations can only be guaranteed if the flux‐limited scheme is both implicit and satisfies an additional lower‐bound condition on time‐step size. We show that this condition also applies to standard Galerkin linear finite‐element approximations to the linear diffusion equation. Numerical experiments are provided to demonstrate the behavior of the methods and confirm the theoretical conditions on time‐step size, mesh spacing, and flux limiting for transport problems with and without nonlinear reaction. Copyright © 2003 John Wiley & Sons, Ltd.     

Derivative artificial compression method for conservation laws with multi-dimensional extension

A new resolution-enhancing technique called derivative artificial compression method is developed with multi-dimensional extension. The method is constructed via applying high-resolution difference schemes on derivative equations of conservation laws. In this way, one could overcome the defect of accuracy decay at extreme points that has plagued almost all high-resolution schemes. The new method has high resolution, low dissipation and low diffusion properties, and could enhance the resolution (of numerical solution) both at discontinuities and at extreme points. Numerical experiments are implemented using initial value problems of single conservation law, one-dimensional shock-tube problem, two-dimensional Riemann problems, double Mach reflection problem, and a shock reflection from a wedge. Resolutions of discontinuities, extremes and fine structures are compared between the original TVD scheme, TVD scheme with artificial compression method and TVD scheme with derivative artificial compression method.     

AN ACCURATE FINITE DIFFERENCE SCHEME FOR SOLVING CONVECTION-DOMINATED DIFFUSION EQUATIONS

Approximating convection-dominated diffusion equations requires a very accurate scheme for the convection term. The most famous is the method of backward characteristics, which is very precise when a good interpolation procedure is used. However, this method is difficult to implement in 2D or 3D. The goal of this paper is to show that it is possible to construct finite difference schemes almost as accurate as the method of characteristics. Starting from a family of second- and third- order Lax–Wendroff-type schemes, a TVD and L^∞- stable scheme that is easy to implement in higher dimensions is constructed. Numerical tests are performed on various model problems whose solution is known and on classical problems. Comparisons with some other limiter schemes and the method of characteristics are discussed. © 1997 by John Wiley & Sons, Ltd.     

ON THE CONSTRUCTION OF A THIRD-ORDER ACCURATE MONOTONE CONVECTION SCHEME WITH APPLICATION TO TURBULENT FLOWS IN GENERAL DOMAINS

A formally third-order accurate finite volume upwind scheme which preserves monotonicity is constructed. It is based on a third-order polynomial interpolant in Leonard's normalized variable space. A flux limiter is derived using the fact that there exists a one-to-one map between normalized variable and TVD spaces. This scheme, which is relatively simple and quite compact, is implemented in a staggered general co-ordinates finite volume algorithm including the standard k–ϵ model and applied to the turbulence transport equations. A number of test problems demonstrate the utility of the proposed scheme. It is shown that in cases where turbulence convection is dominant, the application of a higher-order monotone convection scheme to the turbulence equations leads to results which are more accurate than those obtained using the first-order upwind scheme.     

通量限制器对算法性能的影响

在非正交曹线坐标系下,本文给出了求解非线性双曲型Ｅｕｌｅｒ方程的ＬＵ－ＡＵＳＭＬＷ算法。为了改进该算法的性能,将高分辨率ＡＵＳＭＰＷ格式的空间精度由一阶精度扩展到三阶精度。分析了选择通量限制器对算法稳定性、收敛性和精度的影响,并构造了一种新的能量限制器。本文数值结果表明,通量限制器是决定ＬＵ－ＡＵＳＭＰＷ算法性能的关键因素,并且该算法采用本文构造的通量限制器,求解非线性双曲型Ｅｕｌｅｒ方程,具有较     

Approximation of shallow water equations by Roe's Riemann solver

The inviscid shallow water equations provide a genuinely hyperbolic system and all the numerical tools that have been developed for a system of conservation laws can be applied to them. However, this system of equations presents some peculiarities that can be exploited when developing a numerical method based on Roe's Riemann solver and enhanced by a slope limiting of MUSCL type. In the present paper a TVD version of the Lax-Wendroff scheme is used and its performance is shown in 1D and 2D computations. Then two specific difficulties that arise in this context are investigated. The former is the capability of this class of schemes to handle geometric source terms that arise to model the bottom variation. The latter analysis pertains to situations in which strict hyperbolicity is lost, i.e. when two eigenvalues collapse into one.     

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent non-physical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD (CTVD) schemes were tested, one is third-order accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a two-dimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.     

A FULLY NONLINEAR BOUSSINESQ MODELFOR WAVE PROPAGATION BASED ON MUSTA SCHEME

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.     

基于MUSTA格式的全非线性Boussinesq波浪传播数值模型

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.      

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.     

求解双曲型守恒律的半离散三阶中心迎风格式

给出了求解一维双曲型守恒律的一种半离散三阶中心迎风格式,并利用逐维进行计算的方法将格式推广到二维守恒律。构造格式时利用了波传播的单侧局部速度,三阶重构方法的引入保证了格式的精度。时间方向的离散采用三阶TVD Runge—Kutta方法。本文格式保持了中心差分格式简单的优点,即不需用Riemann解算器,避免了进行特征分解过程。用该格式对一维和二维守恒律进行了大量的数值试验,结果表明本文格式是高精度、高分辨率的。     

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.     

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.     

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.