变形法移动网格求解双曲型守恒律方程研究

在现有格式的基础上要提高偏微分方程数值解的分辨率,自适应移动网格技术是一种有效而且可行的方法。文中将文献[1]提出的自适应移动网格技术推广到三角形网格,并将该方法用于求解双曲型守恒量方程。用网格自适应技术求解守恒律问题时,当生成新网格之后,需要将旧网格上的函数值更新到新的网格,并保持物理量的守恒性。针对这个问题,文中提出了函数值更新过程中守恒型插值公式的具体形式,并针对二维双曲型守恒律方程进行了仿真实验,取得了满意的结果。     

三维非结构网格上求解双曲型守恒律方程的一类三阶精度有限体积格式

考虑标量双曲型守恒律方程,对三维非结构四面体网格给出了一类满足局部极值原理的三阶精度有限体积格式.方法的主要思想是时间和空间分开处理,时间离散采用三阶TVD Runge-Kutta方法;对空间,在每一个四面体单元上基于最小二乘原理构造一个二次多项式,结合数值解光滑探测器和梯度限制器,使其在光滑区域具有高阶精度,在间断区域满足局部极值原理.该格式具有间断分辨能力高,编程实现简便,计算速度快等优点.典型算例的数值试验表明,该格式是有效的.     

双曲型守恒律方程的小波解法

本文基于Hamilton-Jacobi方程的小波Galerkin近似和微分算子的小波表示,讨论一维双曲型守恒律方程初值问题的Daubechies小波解．由于小波在空间和时间上的局部性,本方法适用于处理具有奇异解的问题,可以有效的防止数值振荡．数值试验的结果表明,本方法是可行的．     

一种求解双曲守恒律方程的中心型WENO格式

本文发展了一种中心型加权本质无振荡(WENO)格式.该格式通过在原始三阶WENO-JS格式的下风方向增加一个两点候选模板,并将文献[11]中的非线性自适应机制推广到r=2情况,格式记为WENO4-CU.经过近似色散关系分析可以看到,WENO4-CU格式的频谱特性较原始三阶WENO-JS格式具有明显的改进.通过六个典型算例的数值测试表明,WENO4-CU格式在对流动结构的分辨上较原始WENO3-JS、WENO3-M和WENO3-Z格式具有明显提高.     

求解双曲守恒律及其对流扩散方程的中心迎风方法

提出了一种新的求解双曲守恒律方程(组)的四阶半离散中心迎风差分方法．空间导数项的离散采用四阶CWENO(central weighted essentially non—oscillatory)的构造方法,使所得到的新方法在提高精度的同时,具有更高的分辨率．使用该方法产生的数值粘性要比交错的中心格式小,而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小．     

双曲型守恒律的一种三阶松弛格式

对一维双曲型守恒律,给出了一种形式更简单、计算量更小的三阶松弛格式.该格式以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础.由于不用求解Riemann问题和计算非线性通量函数的雅可比矩阵,所以本文格式保持了松弛格式简单的优点.数值试验表明:该方法具有较高的分辨率.     

二维双曲守恒律标量方程的三阶CWENO-型熵相容算法

应用提出的中心加权基本无振荡(CWENO)-型熵相容格式求解了二维双曲守恒律方程初边值问题,对所得数值结果进行了分析与讨论,并通过与准确解的比较发现该数值求解格式稳定性条件可以取到0.6,而激波过渡带只有1~2个网格单元。实验结果表明该数值求解格式分辨率高且数值稳定性好。     

基于PHG平台的非结构四面体网格欧拉方程间断有限元并行求解器

针对计算流体力学对高性能计算的需求,基于三维并行自适应有限元程序开发平台PHG (Par allel Hierarchical Grid)开发了在非结构四面体网格上求解可压缩流欧拉方程的间断有限元法并行求解器(Libdgphg库).该求解器以C++函数库的形式实现数值方法中各项功能.实施了模态基一次间断有限元,采用低耗散的MLP (Multi-dimensional Limiting Process)限制器来抑制间断附近的数值振荡.由于MLP限制器需要所有与当前单元共享顶点的邻近单元的信息,模板较宽,这给程序设计带来一定的困难.我们通过引入辅助向量收集共享顶点的所有单元中的最大、最小单元积分平均值,并归属到单元数据结构上,从而利用PHG内在的通信机制实现MPI分区间的信息交换.通过几个数值算例测试了 Libdgphg库的数值结果以及并行性能.算例表明:该求解器能得到理论精度阶和较高分辨率,同时有良好的并行性能,在千核测试中可达到60%以上的并行效率,可用于流体问题的大规模计算.     

变形Boussinesq方程组的多辛Preissmann格式计算研究

基于Hamilton空间体系的多辛理论,研究了变形Boussinesq方程组的数值解法. 利用Preissman方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律. 数值算例结果表明: 该多辛离散格式具有较好的长时间数值稳定性.     

膜受迫振动方程的多辛格式及其守恒律

基于Hamilton空间体系的多辛理论研究了膜强迫振动问题.利用Runge-Kutta多辛格式构造了一种9×3点半隐式的多辛离散格式,该格式满足多辛守恒律.数值算例结果表明该多辛离散格式不仅能够有效提高数值计算精度,而且能够保持膜振动系统的局部性质.同时利用多辛格式模拟得到的波形图表明多辛方法具有较好的长时间数值稳定性.     

Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws

In this paper, a moving mesh discontinuous Galerkin (DG) method is developed to solve the nonlinear conservation laws. In the mesh adaptation part, two issues have received much attention. One is about the construction of the monitor function which is used to guide the mesh redistribution. In this study, a heuristic posteriori error estimator is used in constructing the monitor function. The second issue is concerned with the solution interpolation which is used to interpolates the numerical solution from the old mesh to the updated mesh. This is done by using a scheme that mimics the DG method for linear conservation laws. Appropriate limiters are used on seriously distorted meshes generated by the moving mesh approach to suppress the numerical oscillations. Numerical results are provided to show the efficiency of the proposed moving mesh DG method.     

H-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the $h$ -adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$ -adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our $h$ -adaptive algorithm using the double tolerance adaptive strategy.     

Mesh Redistribution Strategies and Finite Element Schemes for Hyperbolic Conservation Laws

In this work we consider a new class of Relaxation Finite Element schemes for hyperbolic conservation laws, with more stable behavior on the limit area of the relaxation parameter. Combining this scheme with an efficient adapted spatial redistribution process considered also in this work, we form a robust scheme of controllable resolution. The results on a number of test problems show that this scheme can produce entropic-approximations of high resolution, even on the limit of the relaxation parameter where the scheme lacks of the relaxation mechanism. Thus we experimentally conclude that the proposed spatial redistribution process, has by its own interesting stabilization properties for computational solutions of conservation law problems.     

Method of Moving Frames to Solve Conservation Laws on Curved Surfaces

A new numerical framework is proposed to solve partial differential equations on curved surfaces by using the orthogonal moving frames at each grid point to compute the gradient of a scalar variable. We call this framework the method of moving frames (MMF) that is adopted and modified from the works of é. Cartan. Compared to the Eulerian method and the Lagrangian multiplier method, the MMF method uses only the surface as the domain, not additionally the ambient space enclosing it. Also different from directly solving the equations with respect to the curved axis, the MMF method is free of the metric tensors. This uniqueness is the consequence of the virtual and penalty extension of the variables in a special direction, called the exponential direction, instead of the surface normal direction that is typically taken. The exponential extension eliminates the need to extend the computational domain and the variables outside the curved surfaces, but the variables outside the curved surfaces are not extended in the direction of the surface normal, yielding an extension error. However, the overall error for the MMF scheme, caused by the extension error, is of high order in L ₂ error with respect to space discretization. This high convergence rate implies that the exponential error can be made negligible compared to the error of differentiation and integration, which are also expressed with space discretization but with lower order, in adaptively-refined meshes proportional to the Gaussian curvature. As the first application of the MMF method, conservation laws are considered on curved surfaces. To display the exponential convergence and the unique features of the MMF scheme, convergence tests are demonstrated on four different types of surfaces: an open spherical shell, a closed spherical shell, an irregular closed surface, and a non-convex closed surface.     

Multirate Timestepping Methods for Hyperbolic Conservation Laws

This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps (restricted by the largest value of the Courant number on the grid) and therefore results in more efficient algorithms. Numerical results obtained for the advection and Burgers’ equations confirm the theoretical findings. This work was supported by the National Science Foundation through award NSF CCF-0515170.     

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge–Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax–Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax–Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge–Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.     

Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws

In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to $(2k+m)$ -th order in the negative-order norm, where $m$ depends upon the chosen flux.     

Relaxed High Resolution Schemes for Hyperbolic Conservation Laws

Relaxed, essentially non-oscillating schemes for nonlinear conservation laws are presented. Exploiting the relaxation approximation, it is possible to avoid the nonlinear Riemann problem, characteristic decompositions, and staggered grids. Nevertheless, convergence rates up to fourth order are observed numerically. Furthermore, a relaxed, piecewise hyperbolic scheme with artificial compression is constructed. Third order accuracy of this method is proved. Numerical results for two-dimensional Riemann problems in gas dynamics are presented. Finally, the relation to central schemes is discussed.