一类高精度TVD差分格式及其应用

构造了一维非线性双曲型守恒律的一个新的高精度、高分辨率的守恒型TvD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，重构各细小区间交界面上的状态变量，并加以校正；其次，利用近似Riemann解计算细小区间交界面上的数值通量，并结合高阶Runge—Kutta 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TVD特性。该格式适合于使用分量形式计算而无须进行局部特征分解。通过计算几个典型的问题，验证了格式具有高精度、高分辨率且计算简单的优点。     

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。     

二维双曲型守恒律的一种五阶松弛格式

松弛格式是Jin和Xin提出的无振荡有限差分方法,其主要思想是将守恒律转化为松弛方程组进行求解.本文用逐维五阶WENO重构和显隐式Runge-Kutta方法对松弛方程组的空间和时间进行离散,得到了一种求解二维双曲型守恒律五阶松弛格式.所得格式保持了松弛格式简单的优点,不用求解Riemann问题和计算通量函数的雅可比矩阵.通过二维Burgers方程和二维浅水方程的数值算例验证了格式的有效性.     

求解双曲型守恒律的半离散中心差分格式

给出了一种求解双曲型守恒律的三阶半离散中心差分格式。该格式以一种推广的三阶重构为基础,同时考虑了波传播的局部速度。格式的构造方法是利用重构,先计算非一致交错网格上的均值,再将该网格均值投影回原来的非交错网格,得到新的全离散中心差分格式,该格式有半离散形式。本文半离散格式保持了中心差分格式简单的优点,即不需用R iemann解算器,避免了进行特征解耦。它具有守恒形式,数值通量满足相容性条件。数值试验结果表明该格式是高精度、高分辨率的。     

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。     

求解多维双曲守恒律方程组的四阶半离散格式

提出了求解多维双曲守恒律方程组的四阶半离散格式。该方法以中心加权基本无振荡(CWENO)重构为基础,同时考虑到在R iemann扇内波传播的局部速度,从而回避了计算过程中的网格交错,建立了数值耗散较小的介于迎风格式和中心格式之间的半离散格式。本文的四阶半离散格式是Kurganov等人的三阶半离散格式的高阶推广。大量的数值算例充分说明了本文方法的高分辨率和稳定性。     

一类格心型ALE有限体积格式方法

现在国内外流行的ALE有限体积格式基本上都基于交错网榕进行格式的离散.该类格武在进行重映时,速度、密度和能量需要分别进行重映计算,效率较低,而且由于速度在网格角点.而密度、能量在网格中心,重映时会出现动能和内能不协调现泉.本文在巳有格心型Lagrange有限体积格式研究的基础上,结合Abgrall R.等关于榕心型格式下的网格角点速度的计算方法,利用最小二乘法进行高阶插值多项式重构,构造了一类新的格心型的高精度Lagrangian有限体积格式,并结合有效的高精度ENO守恒重映方法,获得了一类格心型的高精度ALE有限体积格式.数值试验的结果说明本文的格式是有效的,高精度的,收敛的,并且避免了物理量的不协调现象.     

拉格朗日高阶中心型守恒气体动力学格式

提出了拉格朗日高阶中心型守恒气体动力学格式。用产生于当前时刻子网格密度和当前时刻网格声速的子网格压力构造了子网格力,用加权本质无震荡方法构造的高阶子网格力构造了高阶空间通量,借助时间中点通量的泰勒展开完成了高阶时间通量离散,利用动量守恒条件使得格点速度以与网格面的数值通量相容的方式计算。编制了拉格朗日高阶中心型守恒气体动力学格式,对Saltzman活塞问题进行了数值模拟,数值结果表明,拉格朗日高阶中心型守恒气体动力学格式的有效性和精确性.     

二维溃坝绕流的WENO格式数值模拟

WENO(Weighted essentially non-oscillatory scheme)格式是一类新的高精度无振荡差分格式.本文将WENO格式和Runge-Kutta时间离散的思想应用于二维浅水方程组的求解,数值模拟矩形河道中大坝瞬间局部溃倒,下游有障碍物的洪水演进过程,并对模拟结果进行了分析,表明采用WENO格式所建立的高分辨率模型能够有效地模拟溃坝波的演进过程.     

基于Level Set的多维双曲守恒律标量方程的激波追踪方法

结合四阶CWENO(Cemral Weighted Essentially Non-Oscillatory)格式、四阶NCE(Natural Continuous Extensions)Runge-Kutta法和Level Set方法，很好地处理了一维双曲守恒律标量方程的激波追踪问题。针对二维双曲守恒律标量方程，成功地用五阶WENO格式、非TVD格式的四阶Runge-Kutta方法和Level Set方法进行激波追踪。将所得的数值解与标准的高阶激波捕捉方法所得的数值解进行比较，说明基于Level Set的激波追踪方法的有效性与逐点收敛性。     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.      

Shock‐capturing with natural high‐frequency oscillations

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock‐capturing under the influence of natural high‐frequency oscillations. The conjugate low‐ and high‐pass filters are constructed based on the principle of the discrete singular convolution (DSC), a local spectral method. The accuracy and resolution of the DSC basic algorithm are accessed with a one‐dimensional advection equation. Two Euler systems, the advection of an isotropic vortex flow and the interaction of shock–entropy wave are utilized to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock–entropy wave interaction. Numerical experiments indicate that the CFOR scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 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.     

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.     

High-order Lagrangian cell-centered conservative scheme on unstructured meshes

A high-order Lagrangian cell-centered conservative gas dynamics scheme is presented on unstructured meshes. A high-order piecewise pressure of the cell is intro- duced. With the high-order piecewise pressure of the cell, the high-order spatial discretiza- tion fluxes are constructed. The time discretization of the spatial fluxes is performed by means of the Taylor expansions of the spatial discretization fluxes. The vertex velocities are evaluated in a consistent manner due to an original solver located at the nodes by means of momentum conservation. Many numerical tests are presented to demonstrate the robustness and the accuracy of the scheme.     

On the accuracy of high-order discretizations for underresolved turbulence simulations

In this paper, we investigate the accuracy of a high-order discontinuous Galerkin discretization for the coarse resolution simulation of turbulent flow. We show that a low-order approximation exhibits unacceptable numerical discretization errors, whereas a naive application of high-order discretizations in those situations is often unstable due to aliasing. Thus, for high-order simulations of underresolved turbulence, proper stabilization is necessary for a successful computation. Two different mechanisms are chosen, and their impact on the accuracy of underresolved high-order computations of turbulent flows is investigated. Results of these approximations for the Taylor–Green Vortex problem are compared to direct numerical simulation results from literature. Our findings show that the superior discretization properties of high-order approximations are retained even for these coarsely resolved computations.