This paper presents an adaptive moving mesh algorithm for two-dimensional (2D) ideal magnetohydrodynamics (MHD) that utilizes a staggered constrained transport technique to keep the magnetic field divergence-free. The algorithm consists of two independent parts: MHD evolution and mesh-redistribution. The first part is a high-resolution, divergencefree, shock-capturing scheme on a fixed quadrangular mesh, while the second part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative-interpolation formula is used to calculate the remapped cell-averages of the mass, momentum, and total energy on the resulting new mesh; the magnetic potential is remapped to the new mesh in a non-conservative way and is reconstructed to give a divergence-free magnetic field on the new mesh. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy, track and resolve strong shock waves in ideal MHD problems, and preserve divergence-free property of the magnetic field. Numerical examples include the smooth Alfve´n wave problem, 2D and 2.5D shock tube problems, two rotor problems, the stringent blast problem, and the cloud-shock interaction problem.

This paper presents a second-order accurate adaptive generalized Riemann problem (GRP) scheme for one and two dimensional compressible fluid flows. The current scheme consists of two independent parts: Mesh redistribution and PDE evolution. The first part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative-interpolation formula is used to calculate the cell-averages and the slopes of conservative variables on the resulting new mesh. The second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the Eulerian GRP scheme, which is directly extended to two dimensional arbitrary quadrilateral meshes. Several numerical examples show that the current adaptive GRP scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points, but also reduces possible errors or oscillations effectively.

It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order O(1) for the high frequency modes unj=λnkeiξj, ξ≈π, but of order O(ξ2) for low frequency modes (ξ≈0). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes.

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.

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.

This paper presents an efficient adaptive mesh redistribution method to solve a non-linear Dirac (NLD) equation. Our algorithm is formed by three parts: the NLD evolution, the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh. At each time level, the equidistribution principle is first employed to iteratively redistribute coarse mesh points, and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity. After an adaptive coarse mesh is generated ideally and finally, each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain, and then the solution vector is remapped on the resulting new fine mesh by an affine method. The NLD equation is finally solved by using a high resolution shock-capturing method on the (fixed) non-uniform fine mesh. Extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third-order rate of convergence, and yields an efficient and fast NLD solver that tracks and resolves both small, local and large solution gradients automatically.

This note gives a numerical investigation for the popular high resolution conservative schemes when applied to inviscid, compressible, perfect gas flows with an initial high density ratio as well as a high pressure ratio. The results show that they work very inefficiently and may give inaccurate numerical results even over a very fine mesh when applied to such a problem. Numerical tests show that increasing the order of accuracy of the numerical schemes does not help much in improving the numerical results. How to cure this difficulty is still open.

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.

Based on a simple projection of the solution increments of the underlying partial differential equations (PDEs) at each local time level, this paper presents a difference scheme for nonlinear Hamilton-Jacobi (H-J) equations with varying time and space grids. The scheme is of good consistency and monotone under a local CFL-type condition. Moreover, one may deduce a conservative local time step scheme similar to Osher and Sanders scheme approximating hyperbolic conservation law (CL) from our scheme according to the close relation between CLs and H-J equations. Second order accurate schemes are constructed by combining the reconstruction technique with a second order accurate Runge-Kutta time discretization scheme or a Lax-Wendroff type method. They keep some good properties of the global time step schemes, including stability and convergence, and can be applied to solve numerically the initial-boundary-value problems of viscous H-J equations. They are also suitable to parallel computing. Numerical errors and the experimental rate of convergence in the Lp-norm, p=1, 2, and ∞, are obtained for several one-and two-dimensional problems. The results show that the present schemes are of higher order accuracy.