TVD格式求解对流扩散方程的最优限制器研究

为探究不同通量限制器应用于TVD(Total Variation Diminishing)格式求解对流扩散方程时的适用性，基于3种典型的TVD格式与10种常用的通量限制器，分别求解了线性对流扩散方程、非线性对流扩散方程、拟线性对流扩散方程。数值结果表明，相比于MUSCL(Monotonic Upstream-centered Scheme for Conservation Laws)和MTVDLF(Modified TVDLF)格式，采用TVDLF(TVD Lax-Friedrichs)格式时，计算结果出现了较为严重的数值耗散；对MUSCL和MTVDLF格式进行具体分析发现，关于阶跃型纯对流问题，Superbee限制器的误差最小，Minmod误差最大。关于高斯型对流扩散问题，Minmod误差最大，Woodward误差最小。而关于阶跃型对流扩散问题及Burgers方程，限制器的类型对实验结果影响并不明显。     

求解复杂交通流模型的低耗散中心迎风格式

针对非均匀道路上的多车种LWR交通流模型,提出一种低耗散中心迎风格式。以4阶中心加权基本无震荡重构和低耗散中心迎风数值通量为基础,通过构造不同形式的全局光滑因子及增大非光滑模板对应的非线性权重优化数值格式的耗散特性,并采用Runge-Kutta方法对半离散数值格式在时间方向上进行离散使其保持4阶精度。对非均匀道路上多车种LWR交通流模型的车道数变化和交通信号灯控制问题进行数值模拟,结果表明该格式具有4阶求解精度,且分辨率高。     

CFD差分格式及限制器计算对比分析

差分格式是计算流体力学中最为核心的因素,一直是CFD发展的主线.为了分析比较各种差分格式和限制器,以激波管Biemann问题为算例,应用八种差分格式和五种限制器进行了计算分析,对比了各种格式对于膨胀波、激波及接触间断的分辨率,讨论了中心型和迎风型格式的粘性机理和优劣,比较了各类限制器的压缩性和耗散性.研究表明,各类差分格式对间断和粘性的处理,是提高格式精度和判别格式优劣的关键.采用MUSCL方法插值时,应权衡压缩性和耗散性,合理选择限制器.各类格式通过与MUSCL高阶插值方法相结合,可以有效提高的计算效率和计算精度.     

求解相对论流体力学方程的低耗散中心迎风格式

针对一维相对论流体力学方程,给出一种数值求解方法.该方法以低耗散中心迎风数值通量为基础,通过分片线性重构来获得空间上的二阶精度,最后采用强稳定龙格库塔方法在时间方向上推进.数值算例验证了该方法的有效性和基本无振荡性.     

三维非定常不可压涡量—速度Navier—Stokes方程组的有限差分法

提出了一种数值求解三维非定常涡量—速度形式的不可压Navier-Stokes方程组的有限差分方法,该方法在空间方向上具有二阶精度,并且系数矩阵具有对角占优性,因此适合高雷诺数问题的数值求解．同时,给出了适合的二阶涡量边界条件．通过对有精确解的狄利克雷边值问题和典型的驱动方腔流问题的数值实验,验证了本文格式的精确性、稳定性和有效性．     

求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

迎风紧致格式与热流计算

１．引 言 众所周知,ＴＶＤ格式是能够高质量地捕捉激波的方法,但在计算粘性绕流时许多ＴＶＤ格式数值耗散太大,不能正确模拟粘性流动,因而无法正确计算热流值．文献［３］指出,采用高精度格式可适当放松对网格雷诺数的要求,因此发展三阶或三阶以上的格式是需要的．文献［４］研究了迎风紧致群速度控制格式（ＵＣＧＶＣ格式）在 Ｅｕｌｅｒ方程中的应用,提高了对激波的分辨率,优于通常二阶精度ＴＶＤ格式．本文在文献［４］的基础上给出了利用迎风紧致格式求解ＮＳ方程．它是ＵＣＧＶＣ格式在粘性流计算中的推广．对于方程中的无粘…     

高精度方法在污染传播中的应用

使用数值模拟的方法研究污染物的传播.通过提高计算方法在一定网格规模下准确分辩和刻画各种尺度流动结构的能力,从而提高小尺度污染传播数值预报的精度.对常用格式的分辨率进行了细致比较,结果表明紧致格式无论在分辨率,还是在模版尺寸上均有较大优势.此外,在模拟带有局部高浓度区的传播问题时,格式的耗散性直接影响计算的准确性和稳定性,常用的差分格式很难得到理想的结果.而改用紧致格式后,取得了比较理想的效果.     

求解非线性Schroedinger方程的一个线性化紧致差分格式

通过对非线性项的局部外推,对非线性Schroedinger方程给出了一个线性化紧致差分格式,运用不动点定理和能量方法证明了格式的唯一可解性,文章还运用能量方法和数学归纳法,避开困难的先验估计,证明格式在空间方向和时间方向分别具有四阶和二阶精度,数值算例验证了格式的精度和数值稳定性．     

一种采用物理模型的实时溃坝动画算法

溃坝水波是一种包含大梯度间断的水流运动,既包含整体的水波运动,又必然夹杂着大量分散的水花.本文采用一种基于物理模型的方法模拟溃坝水波,把溃坝过程看作是含间断波的二维浅水流运动.采用无结构网格的有限体积法求解N-S方程,应用二阶的Reo-MUSCL格式,并采用了适当的限制器,使得我们的模型既不产生过大耗散,又具有较高分辨率.在真实感处理方面,加入了粒子系统以模拟溃坝过程中水花四溅的情形.实验结果表明:本文的方法能够真实有效地模拟溃坝水流的运动.     

Computer evaluation of high order numerical schemes to solve advective transport problems

This study evaluates the performance of three representative high-order finite difference schemes to solve two sets of simple one-dimensional benchmark problems in terms of their ability to resolve spurious oscillation, numerical spreading, and peak clipping. Three models, namely QUICKEST, ULTIMATE, and ENO were constructed to represent the classical high-order schemes without a flux limiter, TVD with a flux limiter, and TVB schemes, respectively. Three sets of results generated by QUICKEST, ULTIMATE, and ENO were compared with the analytical solutions. The first set indicated that none of these high-order schemes could yield satisfactory simulations when the grid size and time-step size specified by the benchmark problems were used. The second set showed that all three numerical schemes generated excellent computations when the grid size was reduced to one-tenth and the time-step size was reduced to one-fifth of those specified by the benchmark problems. The third set demonstrated that the results obtained by these schemes deteriorated even with the reduced grid size and time-step size when 100 folds of simulation times was conducted. The ENO and ULTIMATE schemes successfully eliminated spurious oscillations for all cases as expected. The QUICKEST scheme alleviated the problem of spurious oscillations only when the reduced grid and time-step sizes were used. In terms of numerical spreading and peak clipping, none of the three schemes produced satisfactory results unless the reduced grid and time-step were used. Peak clipping poses a more severe problem for these high order schemes than numerical spreading. A common set of benchmark problems is needed for the evaluation and testing of any numerical scheme.     

Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.     

Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

Simulation of a simple reacting gas flow with an upwind scheme

Good results have been obtained using the Random Choice Method (RCM) in the computation of reacting gas flow problems. The RCM is an unfamiliar method and difficult to program. The question arises as to whether a simpler difference approximation can obtain as effective results with less computational difficulty. Among all difference schemes upwind methods have been proven to have excellent properties. Thus, such methods serve as models for the effectiveness of all difference schemes.A standard upwind scheme modified to include a fractional heat conduction step is used to compute solutions of one dimensional compressible fluid flow equations with a finite heat conduction coefficient. The gas is assumed to be chemically reacting and thus to deposit energy in the field. Comparison is made to the known qualitative behavior of the solutions for different ratios of the reaction rate and the heat conduction coefficient. This difference scheme is seen to compare unfavorably with the RCM.     

欧拉方程的隐式间断有限元算法研究

针对Euler方程,设计了适合间断Galerkin有限元方法的LU-SGS、GMRES以及修正LU-SGS隐式算法。采用Roe通量以及Van Albada限制器技术实现了经典LU-SGS、GMRES算法,引入高阶项误差补偿,发展了修正LU-SGS算法。以NACA0012、RAE2822翼型为例验证分析了算法的可靠性和高效性。结果表明修正LU-SGS算法存储量较少,程序实现方便,而且计算效率是LU-SGS算法的2.5倍以上,接近于循环GMRES算法。     

A numerical study of a particular non-conservative hyperbolic problem

We study the ability of several numerical schemes to solve a non-conservative hyperbolic system arising from a flow simulation of solid-liquid-gas slurries with the so-called virtual mass effect. Two classes of numerical schemes are used: some Roe-type finite volume schemes, which are based on the resolution of linearized Riemann problems, and some (centered or upwind) schemes with an additional artificial diffusion, such as the classical Rusanov scheme. For flow regimes of interest (steady as well as unsteady flows), the computational process breaks down for some schemes. Indeed, for such flows, the system has at least one eigenvalue having a small magnitude in the interior of the computational domain and this is a possible reason for the failure of some upwind schemes using the resolution of a linearized Riemann problem. Such a failure does not appear with, for instance, the Rusanov scheme which is well known for its robustness. Furthermore, since the system is non-conservative, it is not clear what a weak solution is, when the solution is discontinuous (at least, one needs to have the non-conservative equivalent of the Rankine-Hugoniot jump conditions) and we show that the approximate solution given by different numerical schemes converges towards different “weak solutions”.     

一维非定常对流扩散方程的高阶组合紧致迎风格式

通过将对流项采用四五阶组合迎风紧致格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h~4+τ~4).经Fourier精度分析和数值验证,证实了格式的良好性能.三个数值算例包括线性常系数问题,矩形波问题和非线性问题,数值结果表明:该格式具有很高的分辨率,且适用于对高雷诺数问题的数值模拟.     

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.     

An upwind local RBF-DQ method for simulation of inviscid compressible flows

In this paper, an upwind local radial basis function-based differential quadrature (RBF-DQ) scheme is presented for simulation of inviscid compressible flows with shock wave. RBF-DQ is a naturally mesh-free method. The scheme consists of two parts. The first part is to use the local RBF-DQ method to discretize the Euler equation in conservative, differential form on a set of scattered nodes. The second part is to apply the upwind method to evaluate the flux at the mid-point between the reference knot and its supporting knots. The proposed scheme is validated by its application to simulate the supersonic flow in a symmetric, convergent channel and the shock tube problem. The obtained numerical results agree very well with the theoretical data.