双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

间断有限元方法在弹尾超音速喷流计算中的应用

采用间断有限元方法对超音速无粘喷流流动进行数值模拟.将二维双曲守恒方程的间断有限元方法发展到轴对称Euler方程,并就某导弹尾部超音速伴随射流进行数值计算.计算结果与实验照片反映的流动特征吻合较好,与高精度、高分辨率TVD格式的计算结果相比,间断有限元方法的计算结果在轴线反射点附近具有较高的分辨率,表明该方法对激波具有较强的捕捉能力,在激波阵面上不会产生振荡或抹平间断现象.     

Linear high-resolution schemes for hyperbolic conservation laws: TVB numerical evidence

The Osher–Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag–Tang 2D vortex problem. Total-variation-bounded (TVB) behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley–Leverett test simulations.     

Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., [6], [10]. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in [1], [2] can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in [22]. In this paper, we first discuss an extension of the technique in [22], [23], [24] to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in [22]. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.     

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.     

多介质流的高分辨率Euler方法

在多介质流动问题中,不同介质有不同的状态方程。这使通量成为间断函数,从而没有通量的Jacobi矩阵。而用Euler坐标系描述的方程组的很多高分辨率格式都要用到Jacobi矩阵及其特征值和特征向量,即要求通量连续可微。因此必须重新处理整个守恒律方程组。对于γ气体问题将γ看作一个新未知量并增加一个守恒方程,从而使整个方程组的通量成为光滑函数,为高分辨率格式的构造铺平了道路。由于真实流动只遵守三个守恒律,多加的一个守恒律虽然对偏微分方程组没有影响,但对差分方程数值解有影响。这一点在数值实验中已有表现。提出了一个方案将这一影响尽量消除。所用格式可完全照搬单介质流动的任何现有格式。对一维多介质流动Euler方程组的激波管问题的数值实验表明这样处理所构造的格式具有同单介质流动问题同样的效果。     

MUSCL格式TV性质的非线性分析

分析求解非线性双曲型守恒律的MUSCL类格式的TV性质。首先从该类格式的一般形式出发,提出和证明了该类格式实现TVD的需求。所提TVD需求直接表示为对变量变差符号和量值限制,体现了双曲型方程解的依赖域原理,为分析MUSCL格式的TV性质提供了理论工具。同时提出了基于TVD需求再构数值解分布,以降低数值耗散从而提高接触面及膨胀波头/波尾分辨率的基本思路。     

不规则对接网格的分区求解计算

采用一种保持通量守恒的不规则对接网格分区求解中交界面耦合条件的计算方法, 结合有限体积法求解了Euler 方程, 无粘通量取用Van Leer 分裂格式, 构造了一种限制器以实现格式的二阶精度和TVD 性质, 并给出数值算例。     

双曲守恒律的Taylor-Galerkin有限元方法

利用双曲守恒律的Hamilton-Jacobi方程形式,应用Taylor公式与Galerkin有限元给出了求解双曲守恒律的计算方法。采用TVD差分格式的构造思想,对数值通量作修正,在等距网格情形下有限元方法得到的计算格式满足TVD性质,并给出了数值例子。     

Higher Order KFVS Algorithms Using Compact Upwind Difference Operators

A family of high order accurate compact upwind difference operators have been used, together with the split fluxes of the KFVS (kinetic flux vector splitting) scheme to obtain high order semidiscretizations of the 2D Euler equations of inviscid gas dynamics in general coordinates. A TVD multistage Runge–Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic flow problems which indicate the higher accuracy and low diffusion realizable in such schemes.     

Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms

In [16], [17], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions), it is more difficult to design high order schemes which do not produce negative density or pressure. In this paper, we first show that our framework to construct positivity-preserving high order schemes in [16], [17] can also be applied to Euler equations with a general equation of state. Then we discuss an extension to Euler equations with source terms. Numerical tests of the third order Runge–Kutta DG (RKDG) method for Euler equations with different types of source terms are reported.     

Splitting based finite volume schemes for ideal MHD equations

We design finite volume schemes for the equations of ideal magnetohydrodynamics (MHD) and based on splitting these equations into a fluid part and a magnetic induction part. The fluid part leads to an extended Euler system with magnetic forces as source terms. This set of equations are approximated by suitable two- and three-wave HLL solvers. The magnetic part is modeled by the magnetic induction equations which are approximated using stable upwind schemes devised in a recent paper [F. Fuchs, K.H. Karlsen, S. Mishra, N.H. Risebro, Stable upwind schemes for the Magnetic Induction equation. Math. Model. Num. Anal., Available on conservation laws preprint server, submitted for publication, URL: <http://www.math.ntnu.no/conservation/2007/029.html>]. These two sets of schemes can be combined either component by component, or by using an operator splitting procedure to obtain a finite volume scheme for the MHD equations. The resulting schemes are simple to design and implement. These schemes are compared with existing HLL type and Roe type schemes for MHD equations in a series of numerical experiments. These tests reveal that the proposed schemes are robust and have a greater numerical resolution than HLL type solvers, particularly in several space dimensions. In fact, the numerical resolution is comparable to that of the Roe scheme on most test problems with the computational cost being at the level of a HLL type solver. Furthermore, the schemes are remarkably stable even at very fine mesh resolutions and handle the divergence constraint efficiently with low divergence errors.     

高分辨率激波捕捉格式及其CFD应用

在曲线坐标下将高分辨率AUSMPW激波捕捉格式扩展到三维问题。为了提高计算精度,采用三阶MUSCL格式,将AUSMPW格式与隐械时间推进法LU－SGS方法结合,应用于可压缩无粘和有粘流动的数散模拟,计算结果与文献中计算结果和实验数据相符,并将该格式与CD、TVD、FVS和FD格式进行了比较。     

基于静动态重构的高阶DG/FV混合格式在二维非结构网格中的推广

通过比较间断Galerkin有限元方法(DGM)和有限体积方法(FVM),提出"静态重构"和"动态重构"的概念,进一步建立基于静动态"混合重构"算法的三阶DG/FV混合格式.在DG/FV混合格式中,单元平均值和一阶导数由DGM方法"动态重构",二阶导数利用FVM方法"静态重构";在此基础上,构造高阶多项式插值函数,得到...     

On large time step TVD scheme for hyperbolic conservation laws and its efficiency evaluation

A large time step (LTS) TVD scheme originally proposed by Harten is modified and further developed in the present paper and applied to Euler equations in multidimensional problems. By firstly revealing the drawbacks of Harten’s original LTS TVD scheme, and reasoning the occurrence of the spurious oscillations, a modified formulation of its characteristic transformation is proposed and a high resolution, strongly robust LTS TVD scheme is formulated. The modified scheme is proven to be capable of taking larger number of time steps than the original one. Following the modified strategy, the LTS TVD schemes for Yee’s upwind TVD scheme and Yee–Roe–Davis’s symmetric TVD scheme are constructed. The family of the LTS schemes is then extended to multidimensional by time splitting procedure, and the associated boundary condition treatment suitable for the LTS scheme is also imposed. The numerical experiments on Sod’s shock tube problem, inviscid flows over NACA0012 airfoil and ONERA M6 wing are performed to validate the developed schemes. Computational efficiencies for the respective schemes under different CFL numbers are also evaluated and compared. The results reveal that the improvement is sizable as compared to the respective single time step schemes, especially for the CFL number ranging from 1.0 to 4.0.     

A higher-order macroscopic model for pedestrian flows

This paper develops a higher-order macroscopic model of pedestrian crowd dynamics derived from fluid dynamics that consists of two-dimensional Euler equations with relaxation. The desired directional motion of pedestrians is determined by an Eikonal-type equation, which describes a problem that minimizes the instantaneous total walking cost from origin to destination. A linear stability analysis of the model demonstrates its ability to describe traffic instability in crowd flows. The algorithm to solve the macroscopic model is composed of a splitting technique introduced to treat the relaxation terms, a second-order positivity-preserving central-upwind scheme for hyperbolic conservation laws, and a fast-sweeping method for the Eikonal-type equation on unstructured meshes. To test the applicability of the model, we study a challenging pedestrian crowd flow problem of the presence of an obstruction in a two-dimensional continuous walking facility. The numerical results indicate the rationality of the model and the effectiveness of the computational algorithm in predicting the flux or density distribution and the macroscopic behavior of the pedestrian crowd flow. The simulation results are compared with those obtained by the two-dimensional Lighthill-Whitham-Richards pedestrian flow model with various model parameters, which further shows that the macroscopic model is able to correctly describe complex phenomena such as “stop-and-go waves” observed in empirical pedestrian flows.     

Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations

The goal of this work is to extend finite volume WENO and central WENO schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function. In particular, we apply proposed schemes to the shallow water and the open-channel flow equations where the source term depends on the channel geometry. For obtaining stable numerical schemes that are free of spurious oscillations, it becomes crucial to use the decomposed source term evaluation, which maintains the balancing between the flux gradient and the source term. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the source term decomposition, in combination with the Roe approximate Riemann solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property – the property of preserving the quiescent flow exactly. When the central WENO schemes are applied, additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes, as well as with some other first- and second-order numerical schemes.     

Positivity-preserving high order finite difference WENO schemes for compressible Euler equations

In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.     

各向异性扩散问题Kershaw格式的守恒保正修复算法

针对各向异性扩散方程Kershaw格式的数值解在正交网格及扭曲网格上计算出负的现象,给出一种守恒的保正修复算法（CENZ）,该算法对简单遇负置零（ENZ）方法进行改进,使修复后的数值解不仅具有非负性,而且保持法向通量的局部守恒性.数值算例表明,该方法不受计算网格类型和扩散系数各向异性比的限制,可用于对任意违背单调性（或保正性）的有限体积格式数值解的修复.     

On maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.