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.
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.
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.
