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.

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.

This paper presents theoretical and numerical analyses of the sonic point glitch based on some numerical schemes for the Burgers equation and the Euler equations in fluid mechanics. The sonic glitch is formed in the sonic rarefaction fan. It has no any direct connection with the violation of the entropy condition or the size of numerical viscosity of a finitedifference scheme. Our results show that it is mainly coming from a disparity in wave speeds across the sonic point. If numerical viscosity depends on the characteristic direction, then the disparity may be formed between the numerical and physical wave speeds around the sonic point, and triggers the sonic wiggle in the numerical solution. We also find that the initial data reconstruction technique of van Leer can effectively eliminate the flaw around the sonic point for the Burgers equation. Some other possible cures are also suggested.

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.

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.