一维双曲守恒方程组的Taylor-Galerkin有限元方法

１．引言 在文章［８］中,利用双曲守恒律的Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式,应用 Ｇａｌｅｒｋｉｎ有限元给出了求解一维双曲守恒律的计算方法．不同于间断有限元方法［２］、［３］和 Ｔａｙｌｏｒ－Ｇａｌｅｒｋｉｎ有限元方法［１］求解双曲守恒律,文章［８］采用连续 Ｇａｌｅｒｋｉｎ有限元求解双曲守恒律． 在文章［８］中,对差分方法和有限元方法求解双曲守恒律作了较为详细的讨论．同时在文章［８］中,采用积分变换,将双曲守恒律方程变成 Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式．由于 Ｈａｍｉｌｔｏｎ－Ｊａｃｏ…     

Diffusive Regularization Inverse PINN Solutions to Discontinuous Problems北大核心CSCD

双曲守恒律方程间断问题的求解是该类方程数值求解问题研究的重点之一.采用PINN (physics-informed neural networks)求解双曲守恒律方程正问题时需要添加扩散项,但扩散项的系数很难确定,需要通过试算方法来得到,造成很大的计算浪费.为了捕捉间断并节约计算成本,对方程进行了扩散正则化处理,将正则化方程纳入损失函数中,使用守恒律方程的精确解或参考解作为训练集,学习出扩散系数,进而预测出不同时刻的解.该算法与PINN求解正问题方法相比,间断解的分辨率得到了提高,且避免了多次试算系数的麻烦.最后,通过一维和二维数值试验验证了算法的可行性,数值结果表明新算法捕捉间断能力更强、无伪振荡和抹平现象的产生,且所学习出的扩散系数为传统数值求解格式构造提供了依据.     

非线性双曲守恒律方程基于偏迎风数值通量RKDG方法的最优误差估计

针对具有光滑解的一维非线性双曲守恒律方程,研究了Runge-Kutta间断Galerkin (RKDG)方法,其中空间变量采用基于偏迎风数值通量的间断Galerkin方法,时间变量采用三阶显式全变差不增的Runge-Kutta方法.借助能量技术以及最新提出的广义Gauss-Radau投影,证明了通常时空限制条件下全离散方法的最优误差估计.数值实验验证了理论结果.     

一个求解多维守恒律方程组的二阶显式有限元格式

１．引言 近年来,在非结构网格上求解双曲型守恒律的数值方法引起了较为广泛的关注,出现了有限体积方法［１］,间断 Ｇａｌｅｒｋｉｎ方法 ［２］,流线扩散方法［３］,以及 ＮＮＤ格式 ［４］等．我们在［６,７］中提出了一种求解双曲型守恒律方程式的有限元方法,它是在一个求解对流扩散问题的有限元方法 ［５］的基础上发展起来的．它是一个显式有限元方法,因此计算量很小．在这个方法中,我们将任意维的问题归结为在单元棱边上的一维计算,引入了积分因子,因此在单元内部可以容纳边界层．这样,它特别适合于对流占优问题以及双曲…     

用WENO方法求解双曲型守恒律方程组的初(边)值问题

本文用WENO算法解决双曲型守恒律方程组初（边值）问题．给出一种满足熵条件、Sδ熵条件和边界熵条件的WENO算法．通过这个算法就能得到守恒律方程组的数值解，数值解和理论解是非常吻合的．     

基于不同数值流通量求解可压缩Euler方程组的Lax-Wendroff控制体积方法（英文）

基于Lax-Wendroff时间离散的控制体积间断Petrov-Galerkin方法是求解双曲守恒律的一种高精度和高分辨率数值方法.本文通过几个数值算例对8种数值流通量的数值表现作了详尽的比较,内容涉及耗时、精度、分辨率以及模拟复杂波形相互作用的能力.     

求解带有移动界面的线性对流方程的浸入界面有限体积法（英文）

本文为一类带有移动界面的守恒律方程提出了耦合高分辨率格式的数值算法.这种算法是在一致大小的笛卡尔网格上导出而满足标准的双曲型稳定条件.文末列举数值算例研究这种算法的收敛性和数值精度.     

双曲守恒律的几种新数值方法的比较研究

本文就一维线性双曲方程的光滑和间断两种初值问题的求解,对双曲守恒律的三种新数值方法,即,WENO方法、间断Galerkin方法和全局复合方法,进行了数值比较实验,在精度、计算速度等方面的比较上,对这三个方法有了一个较详细的了解,得到了一些有用的结论。     

求解二相LWR交通流模型的低耗散中心迎风格式

将求解双曲型守恒律方程的低耗散中心迎风格式和5阶WENO-ZQ格式相结合,推广应用于求解二相LWR交通流模型方程.并在时间方向上推进采用具有强稳定性的4阶Rung-Kutta方法.最后结合Riemann问题及现实生活所遇到的交通流现象进行设计和分析.通过数值算例证明该格式具较强的稳定性和较高的精度,得到了令人满意的结果.     

具非齐项的守恒律形式双曲方程组解的唯一性

关于守恒律形式的真正非线性双曲方程组解的唯一性问题,R.J.DiPerna证明了分片Lipschitz连续解在满足熵条件的有界BV解类中是唯一的,并且对Lipschitz连续解还得到了在上述解类中的L~2稳定性(注意,对间断解来说,L~2稳定性一般不成立),本文首先对具非齐项(可为非线性项)的守恒律形双曲组定义熵一熵流量以及相应的熵条件,然后把[1]的主要结果推广到这种具非齐项的守恒律形双曲组的情形。     

A smoothness/shock indicator for the RKDG methods

A smoothness/shock indicator is proposed for the RKDG methods solving nonlinear conservation laws. A few numerical experiments are presented as evidence that the indicator helps in detecting shocks, high order discontinuities, regions of smooth solutions, and numerical “instability”.     

An h-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws

In Zhu and Qiu (J Comput Phys 228:6957–6976, 2009), we systematically investigated adaptive Runge-Kutta discontinuous Galerkin (RKDG) methods for hyperbolic conservation laws with different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance for adaptive computation to save computational cost. In this follow-up paper, we extend the method to solve two-dimensional problems. Although the main idea of the method for two-dimensional case is similar to that for one-dimensional case, the extension of the implementation of the method to two-dimensional case is nontrivial because of the complexity of the adaptive mesh with hanging nodes. We lay our emphasis on the implementation details including adaptive procedure, solution projection, solution reconstruction and troubled-cell indicator. Extensive numerical experiments are presented to show the effectiveness of the method.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

大密度比和大压力比可压缩流的数值计算

将WENO方法、RKDG方法、RKDG方法结合原来的Ghost Fluid方法以及RKDG方法结合改进的Ghost Fluid方法,应用到大密度比和大压力比的单相流以及气-气、气-液两相流的数值计算,并对计算结果进行了比较分析．结果表明,与其它的方法相比,RKDG方法结合改进的Ghost Fluid方法得到了高分辨率的计算结果,可以捕捉到正确的激波位置,随着网格的加密,计算解收敛到物理解．     

Subsymmetries and Their Properties

We introduce a subsymmetry of a differential system as an infinitesimal transformation of a subset of the system that leaves the subset invariant on the solution set of the entire system. We discuss the geometric meaning and properties of subsymmetries and also an algorithm for finding subsymmetries of a system. We show that a subsymmetry is a significantly more powerful tool than a regular symmetry with regard to deformation of conservation laws. We demonstrate that all lower conservation laws of the nonlinear telegraph system can be generated by system subsymmetries.     

Effective solving of three-dimensional gas dynamics problems with the Runge-Kutta discontinuous Galerkin method

In this paper we present the Runge-Kutta discontinuous Galerkin method (RKDG method) for the numerical solution of the Euler equations of gas dynamics. The method is being tested on a series of Riemann problems in the one-dimensional case. For the implementation of the method in the three-dimensional case, a DiamondTorre algorithm is proposed. It belongs to the class of the locally recursive non-locally asynchronous algorithms (LRnLA). With the help of this algorithm a significant increase of speed of calculations is achieved. As an example of the three-dimensional computing, a problem of the interaction of a bubble with a shock wave is considered.     

Adaptive Runge-Kutta Discontinuous Galerkin Methods with Modified Ghost Fluid Method for Simulating the Compressible Two-Medium Flow

In this paper, we investigate using the adaptive Runge-Kutta discontinuous Galerkin (RKDG) methods with the modified ghost fluid method (MGFM) in conjunction with the adaptive RKDG methods for solving the level set function to simulate the compressible two-medium flow in one and two dimensions. A shock detection technique (KXRCF method) is adopted as an indicator to identify the troubled cell, which serves for further numerical limiting procedure which uses a modified TVB limiter to reconstruct different degrees of freedom and an adaptive mesh refinement procedure. If the computational mesh should be refined or coarsened, and the detail of the implementation algorithm is presented on how to modulate the hanging nodes and redefine the numerical solutions of the two-medium flow and the level set function on such adaptive mesh. Extensive numerical tests are provided to illustrate the proposed adaptive methods may possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow region and material interfacial vicinities of the two-medium flow region.     

Nonlinear self-adjointness through differential substitutions

It is known (Ibragimov, 2011; Galiakberova and Ibragimov, 2013) [14,18] that the property of nonlinear self-adjointness allows to associate conservation laws of the equations under study, with their symmetries. In this paper we show that, even when the equation is nonlinearly self-adjoint with a non differential substitution, finding the explicit form of the differential substitution can provide new conservation laws associated to its symmetries. By using the general theorem on conservation laws (Ibragimov, 2007) [11] and the property of nonlinear self-adjointness we find some new conservation laws for the modified Harry-Dym equation. By using a differential substitution we construct a conservation law for the Harry-Dym equation, which has not been derived before using Ibragimov method.     

Symmetry analysis, conservation laws of a time fractional fifth-order Sawada-Kotera equation

In this paper, we intend to study the symmetry properties and conservation laws of a time fractional fifth-order Sawada-Kotera (S-K) equation with Riemann-Liouville derivative. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation. Based on this, we find that the S-K equation can be reduced to a fractional ordinary differential equation with Erdelyi-Kober derivative by the similarity variable and transformation. Furthermore, we construct some conservation laws for the S-K equation using the idea in the Ibragimov theorem on conservation laws and the fractional generalization of the Noether operators.     

一种守恒型间断跟踪法在一维单守恒律方程上的程序实现

茅在近几年发展了一种守恒型的间断跟踪法（见[6],[7]），该跟踪法是以解的守恒性质作为跟踪的机制而不是传统的跟踪法利用Rankine-Hugoniot条件。本文的目的是对该算法在一维单守恒律的情况进行程序实现，做成一个对任意初值问题都适应的强健的算法，可处理任意的间断相互作用。在文章的第三节给出了一个数值算例，并与用ENO格式（见[8]所算得的结果进行比较。