This paper presents a further numerical study of the interaction dynamics for solitary waves in a nonlinear Dirac model with scalar self-interaction,the Soler model,by using a fourth order accurate Runge-Kutta discontinuous Galerkin method.The phase plane method is employed for the first time to analyze the interaction of Dirac solitary waves and reveals that the relative phase of those waves may vary with the interaction.In general,the interaction of Dirac solitary waves depends on the initial phase shift.If two equal solitary waves are in-phase or out-of-phase initially,so are they during the interaction;if the initial phase shift is far away from 0 andπ,the relative phase begins to periodically evolve after a finite time.In the interaction of out-of-phase Dirac solitary waves,we can observe:(a)full repulsion in binary and ternary collisions,depending on the distance between initial waves;(b)repulsing first,attracting afterwards,and then collapse in binary and ternary collisions of initially resting two-humped waves;(c)one-overlap interaction and two-overlap interaction in ternary collisions of initially resting waves.
In this paper, two new nonconforming hexagonal elements are presented, which are based on the trilinear function space Q1^（3） and are edge-oriented, analogical to the case of the rotated Q1 quadrilateral element. A priori error estimates are given to show that the new elements achieve first-order accuracy in the energy norm and second-order accuracy in the L^2 norm. This theoretical result is confirmed by the numerical tests.
This paper is concerned with the adaptive grid method for computations of the Euler equations in fluid dynamics.The new feature of the present moving mesh algorithm is the use of a dimensional-splitting type monitor function,which is to increase grid concentration in regions containing shock waves and contact discontinuities or their interactions.Several two–dimensional flow problems are computed to demonstrate the effectiveness of the present adaptive grid algorithm.
An a posteriori error estimator is obtained for a nonconforming finite element approximation of a linear elliptic problem, which is derived from a corresponding unbounded domain problem by applying a nonlocal approximate artificial boundary condition. Our method can be easily extended to obtain a class of a posteriori error estimators for various conforming and nonconforming finite element approximations of problems with different artificial boundary conditions. The reliability and efficiency of our a posteriori error estimator are rigorously proved and are verified by numerical examples.
This paper presents a class of high resolution local time step schemes for nonlinear hyperbolic conservation laws and the closely related convection-diffusion equations, by projecting the solution increments of the underlying partial differential equations （PDE） at each local time step. The main advantages are that they are of good consistency, and it is convenient to implement them. The schemes are L^∞ stable, satisfy a cell entropy inequality, and may be extended to the initial boundary value problem of general unsteady PDEs with higher-order spatial derivatives. The high resolution schemes are given by combining the reconstruction technique with a second order TVD Runge-Kutta scheme or a Lax-Wendroff type method, respectively. The schemes are used to solve a linear convection-diffusion equation, the nonlinear inviscid Burgers＇ equation, the one- and two-dimensional compressible Euler equations, and the two-dimensional incompressible Navier-Stokes equations. The numerical results show that the schemes are of higher-order accuracy, and efficient in saving computational cost, especially, for the case of combining the present schemes with the adaptive mesh method [15]. The correct locations of the slow moving or stronger discontinuities are also obtained, although the schemes are slightly nonconservative.
研究了在Burgers方程跨声速稀疏波计算中遇到的sonic point glitch问题,对它产生的原因及其与数值格式熵条件的关系进行了分析.对若干著名格式,按照是否满足熵条件进行了分类.为了消除sonic point glitch现象,提出了一种新的两步分裂方法,并用这种方法改进了一系列典型格式.数值试验表明这是一种很好的消除sonic point glitch的方法.
在二维散乱离散点集上研究一类无网格方法——有限点方法(Finite Point Method,简称FPM),建立方法的基础.采用方向微商和方向差商讨论有限点方法,建立各阶各方向微商间的关系式.利用这些关系式,根据被逼近点的邻点数目差异,分别建立数值方向微商的五点公式及少点(两点、三点、四点)公式;研究五点公式的可解性条件与可允许邻点集;获得典型微分算子的数值方向微商公式等.理论分析和数值试验表明,随着邻点数目的增加,相应数值公式的逼近精度随之提高.这类近似公式不仅为在散乱离散点集上构造各类偏微分方程的格式奠定了基础,同时,也可应用于偏微分方程非结构网格计算方法,提高方法的精度.
