An AMR(adaptive mesh refinement) method is presented with different time steps in the cells at inequable levels. In order to get high accuracy in the cells on the interlace, the AMR method is adopted together with the level set method and the ghost fluid method. Euler equations is solved by a WENO algorithm which is easily realized while getting a high resolution. Compared with the method without considering AMR or the level set method, our method shows an advantage in achieving high resolution on the interface. The result indicates that these methods work well together in the numerical simulation.

Based on the velocity remapping strategy of the SALE algorithm, three methods are proposed to improve the accuracy and monotonicity of momentum flux calculation on a 1D staggered grid: the SUR method which employs the upwind-type slope and a repairing procedure; the SM method which employs a minmod slope limiter; and the SLR method which employs a new slope limiter and a repairing procedure. These methods are second-ordered, conservative and bound-preserving. They inherit advantages of simplicity and high efficiency of the SALE algorithm. They are easily generalized to the two-dimensional case.