This Letter presents a numerical study of the interaction dynamics for the solitary waves of a nonlinear Dirac field with scalar self-interaction by using a fourth order accurate Runge-Kutta discontinuous Galerkin (RKDG) method. Some new interaction phenomena are observed: (a) a new quasi-stable long-lived oscillating bound state from the binary collisions of a single-humped soliton and a two-humped soliton; (b) collapse in binary and ternary collisions; (c) strongly inelastic interaction in ternary collisions; and (d) bound states with a short or long lifetime from ternary collisions.

This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a half-staggered, moving quadrilateral mesh to keep the velocity field divergence-free, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and mesh-redistribution. In the first part, the incompressible Navier-Stokes equations are solved on a fixed half-staggered mesh by the rotational incremental pressure-correction scheme, and the Allen-Cahn type of phase equation is approximated by a conservative, second-order accurate central difference scheme, where the Lagrangian multiplier is used to preserve the mass-conservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the high-resolution conservative interpolation, while the non-conservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergence-free velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.

An adaptive ghost fluid-nite volume method is developed for one-and two-dimensional compressible multi-medium flows in this work. It couples the real ghost fluid method (GFM) [SIAM J. Sci. Comput. 28 (2006) 278] and the adaptive moving mesh method [SIAM J. Numer. Anal. 41(2003) 487; J. Comput. Phys. 188(2003) 543], and thus retains their advantages. This work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the GFM around the material interface and other discontinuities. Besides the improvement of flow field resolution, the adaptive ghost fluid method also largely increases the computational efficiency. Several numerical experiments are conducted to demonstrate robustness and efficiency of the current method. They include several 1D and 2D gas-water flow problems, involving a large density gradient at the material interface and strong shock-interface interactions. The results show that our algorithm can capture the shock waves and the material interface accurately, and is stable and robust even solution with large density and pressure gradients.

This paper presents a second-order accurate adaptive Godunov method for twodimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41: 487-515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometrybased conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.

First-order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a (2K+1)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.