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.

We develop efficient moving mesh algorithms for one-and two-dimensional hyperbolic systems of conservation laws.The algorithms are formed by two independent parts: PDE evolution and mesh-redistribution.The first part can be any appropriate high-resolution scheme, and the second part is based on an iterative procedure.In each iteration, meshes are first redistributed by an quidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservative-interpolation formula proposed in this work.The iteration for the meshredistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state.The main idea of the proposed method is to keep the mass-conservation of the underlying numerical solution at each redistribution step.In one dimension, we can show that the underlying numerical approximation obtained in the mesh-redistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property.Sev eral test problems in one and two dimensions are computed using the proposed moving mesh algorithm.The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.

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.

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.