Chebyshev谱-Euler混合方法求解一类非线性Burgers方程

将Chebyshev谱方法与Euler方法相结合,对一类非线性Burgers方程进行数值求解,通过数值模拟将其与有限差分法和粒子无网格线混合格式MPS-MAFL方法进行了比较,结果表明这种方法对于求解非线性Burgers方程具有较好的效果.     

求解二维对流扩散方程的投影迭代法

鉴于目前流行的求解大型稀疏代数方程组的投影迭代法中,为提高迭代效率,在迭代前通常需要对稀疏矩阵进行预处理,改善迭代矩阵的条件数,从而减少迭代次数,这使得发展稀疏矩阵的存储技术变得尤为关键。基于二维对流扩散方程的四阶紧致差分格式,将其转化为代数方程组,得到其三对角块形式的系数矩阵,利用稀疏矩阵存储技术和预条件迭代法进行求解,并与传统的中心差分格式所得数值解进行比较,充分说明了方法的高效性和可靠性。     

基于压缩存储技术求解压力Poisson方程的BiCGSTAB算法

基于投影算法所得压力Poisson方程进行数值离散,对离散系统形成的稀疏线性方程组,由于线性方程组的系数矩阵存在大量的零元素,为降低内存存储,本文以一维稀疏存储结构对大规模的系数矩阵进行压缩处理,只存储非零元素。同时,以具有优化性质的BiCGSTAB算法求解压力Poisson方程,显著的提高了计算效率。在相同初始条件下,利用Fortran90完成超松弛迭代法的程序求解压力Poisson方程数值离散所得到的线性方程组进行求解对比。结果表明基于压缩存储的BiCGSTAB算法在求解稀疏线性方程组具有明显的优势,该算法求解速度快、高效、可靠。     

基于异构平台的通量分裂格式性能研究

通量分裂是在方程组条件下实现迎风特性的主要手段,为了实现典型通量分裂格式在CPU/GPU异构平台的性能分析。在NVIDIA GTX1660super上,使用统一设备计算架构(CUDA)编程模型实现一维欧拉求解器;以激波管Riemann问题为算例,对矢通量分裂格式van leer、通量差分分裂格式Roe以及混合通量分裂AUSMPW+进行计算分析;数值结果表明,三种格式在异构计算体系能够得到合理且可用的计算结果;Roe格式激波分辨率最高且在CPU/GPU体系加速效果最好;Van Leer激波分辨率较低于Roe和AUSMPW+,计算效率高但其格式构造中存在大量判断分支,影响了加速性能;AUSMPW+格式激波分辨率与Roe相当,加速性能略好于Van Leer。     

台阶重构的熵格式计算一维Euler方程组

本文设计了一个带三个台阶重构的熵格式,并和带一个台阶的熵格式相结合计算一维Euler方程组.对四个经典的数值算例进行了数值计算,并且与一阶Godunov格式以及二阶的ENO格式进行了数值比较,数值结果表明,本文的格式优于一阶的Godunov格式,和二阶ENO格式相当.     

求解Vlasov-Poisson方程组的一种时间分裂傅里叶谱方法

Vlasov-Poisson方程组是天体物理学和等离子体物理学的一类重要的动力学模型.本文为Vlasov-Poisson方程组设计了一种高效的数值计算方法一时间分裂傅里叶谱方法.在离散该方程组时,我们在时间方向采用时间分裂法,在空间变量方向和速度变量方向均采用傅里叶谱方法.本文首先对一维、二维Vlasov-Poisson方程组的四个守恒量做了分析和证明,然后分别用时间分裂傅里叶谱方法求解一维、二维的Vlasov-Poisson方程组,并给出了详细的算法求解过程.最后通过数值模拟结果证实该方法的准确性和可靠性,并验证了四个守恒量.     

五阶精度WENO差分型格子波尔兹曼算法求解单守恒方程

给出了五阶精度WENO差分型格子波尔兹曼算法求解单守恒模型方程的计算方法.根据WENO差分格式的特点,定义了广义格子波尔兹曼分布函数,将守恒型方程的求解问题,转化成用WENO格式的差分算法对该分布函数进行求解.该方法的意义在于,将高精度高分辨率的WENO格式差分方法与近几十年发展起来的格子波尔兹曼方法相结合,从而很方便地构造出可以用于求解守恒型方程的格子波尔兹曼模型,使格子波尔兹曼方法在可压缩流领域的使用更简单.利用该方法分别构造了不同初值条件下的一维Burgers守恒型方程的求解模型,求出结果,并分析了模型的精度和稳定性.最后总结了方法的优点和不足,以及有待进一步研究解决的问题.     

一维对流扩散反应方程的高精度数值方法

通过将原方程变换为对流扩散方程,将所得方程的对流项采用四阶组合紧致迎风格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的空间半离散格式采用四阶龙格库塔方法进行时间推进,得到了一种求解非定常对流扩散反应问题的高精度方法,其收敛阶为O(h4+τ4).经数值实验并与文献结果进行对比,表明该格式适用于对流占优问题的数值模拟,验证了格式的良好性能.     

Burgers方程的精确解

引入一个变换,将二阶非线性偏微分方程—Burgers方程降阶为一阶的非线性方程,再直接求解该方程,得出了Burgers方程精确解的新形式,并与已有结果完全吻合.这种方法也适合于求解其他非线性偏微分方程.     

基于降阶迭代的振动系统部分特征结构配置北大核心CSCD

采用速度-位移主动控制,提出了一种解决振动系统部分特征结构配置问题的新方法。首先,利用部分特征结构配置的无溢出特性,给出了振动系统速度-位移反馈控制器的参数化表达式。然后,从闭环系统特征关系出发,将部分特征结构配置问题转化为西尔维斯特矩阵方程的求解问题。接着,对西尔维斯特矩阵方程进行降阶处理,并将其转化为等价的低阶矩阵方程组。之后,通过构造新的迭代格式求解矩阵方程组,并由此得到控制器的数值解。最后,进一步给出了迭代格式的收敛性证明及部分特征结构配置问题的求解算法。数值算例验证了所提算法的有效性。     

Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes

We study 2nd-, 4th-, 6th- and 8th-order accurate finite difference schemes approximating systems of conservation laws. Our goal is to utilize the high order of accuracy of the schemes for approximating complicated flow structures and add suitable diffusion operators to capture shocks. We choose appropriate viscosity terms and prove non-linear entropy stability. In the scalar case, entropy stability enables us to prove convergence to the unique entropy solution. Moreover, a limiter function that localizes the effect of the dissipation around discontinuities is derived. The resulting scheme is entropy stable for systems, and also converges to the entropy solution in the scalar case. We present a number of numerical experiments in order to demonstrate the robustness and accuracy of our scheme. The set of examples consists of a moving shock solution to the Burgers’ equation, a solution to the Euler equations that consists of a rarefaction and two contact discontinuities and a shock/entropy wave solution to the Euler equations (Shu’s test problem). Furthermore, we use the limited scheme to compute the solution to the linear advection equation and demonstrate that the limiter quickly vanishes for smooth flows and design/high-order of accuracy is retained. The numerical results in all experiments were very good. We observe a remarkable gain in accuracy when the order of the scheme is increased.     

The influence of the computational mesh on accuracy for initial value problems with discontinuous or nonunique solutions

Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the tine-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.     

Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation

The generalized nonlinear Schrödinger (GNLS) equation is solved numerically by a split-step Fourier method. The first, second and fourth-order versions of the method are presented. A classical problem concerning the motion of a single solitary wave is used to compare the first, second and fourth-order schemes in terms of the accuracy and the computational cost. This numerical experiment shows that the split-step Fourier method provides highly accurate solutions for the GNLS equation and that the fourth-order scheme is computationally more efficient than the first-order and second-order schemes. Furthermore, two test problems concerning the interaction of two solitary waves and an exact solution that blows up in finite time, respectively, are investigated by using the fourth-order split-step scheme and particular attention is paid to the conserved quantities as an indicator of the accuracy. The question how the present numerical results are related to those obtained in the literature is discussed.     

Numerical analysis of homogeneous condensation in rarefaction wave in a shock tube by the space-time CESE method

A detailed numerical investigation has been carried out for the homogeneous condensation of water vapor in a non-stationary rarefaction wave generated in a shock tube. The space-time CESE method has been adopted to simulate the mutual interaction of condensation with the rarefaction wave for an extremely long time. It is found that the homogeneous condensation in the rarefaction wave has a significant influence on the flow due to the latent heat release and the continuous change of cooling rate. Three stages can be defined in this process: the initial stage which contains the onset of the condensation and the formation of the condensation shock waves in both downstream and upstream directions, the oscillating stage which is characterized as the repeat of quench and onset of condensation in the expansion fan approximately in a logarithm time manner, and the asymptotic stage which the oscillating waves are damping out with time and no apparent condensation shock wave is formed.     

A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.     

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley–Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.     

Solution to the Riemann problem in a one-dimensional magnetogasdynamic flow

The Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved approximately. This class of equations includes as a special case the Euler equations of gasdynamics. It has been observed that in contrast to the gasdynamic case, the pressure varies across the contact discontinuity. The iterative procedure is used to find the densities between the left acoustic wave and the right contact discontinuity and between the right contact discontinuity and the right acoustic wave, respectively. All other quantities follow directly throughout the (x, t)-plane, except within rarefaction waves, where an extra iterative procedure is used along with a Gaussian quadrature rule to find particle velocity; indeed, the determination of the particle velocity involves numerical integration when the magneto-acoustic wave is a rarefaction wave. Lastly, we discuss numerical examples and study the solution influenced by the magnetic field.     

Numerical Methods for Euler Equations with Self-similar and Quasi Self-similar Solutions

Some problems of Euler equations have self-similar solutions which can be solved by more accurate method. The current paper proposes two new numerical methods for Euler equations with self-similar and quasi self-similar solutions respectively, which can use existing difference schemes for conservation laws and do not need to redesign specified schemes. Numerical experiments are implemented on one dimensional shock tube problems, two dimensional Riemann problems, shock reflection from a solid wedge, and shock refraction at a gaseous interface. For self-similar equations, one-dimensional results are almost equal to the exact solutions, and two-dimensional results also exhibit considerable high resolution. For quasi self-similar equations, the method can solve solutions that are not but close to self-similar, i.e. quasi self-similar, and this method can also achieve very high resolution when computing time is long enough. Numerical simulations to self-similar and quasi self-similar Euler equations have important implications on the study of self-similar problems, development of high resolution schemes, even the research for exact solutions of Euler equations.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

基于格子Boltzmann方法的一维Burgers方程的数值模拟

基于格子Boltzmann方法(LBM)的一维Burgers方程的数值解法,已有2-bit和4-bit模型。文中通过选择合适的离散速度模型构造出恰当的平衡态分布函数; 然后, 利用单松弛的格子Bhatnagar-Gross-Krook模型、Chapman-Enskog展开和多尺度技术, 提出了用于求解一维Burgers方程的3-bit的格子Boltzmann模型,即D1Q3模型,并进行了数值实验。实验结果表明,该方法的数值解与解析解吻合的程度很好,且误差比现有文献中的误差更小,从而验证了格子Boltzamnn模型的有效性。