In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow bajid of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the CourantFriedrichs-Lewy criterion. Test calculations of one-dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces.The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Ne\iraann and elaborated by other workers in this field.

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We study the nonlinear evolution of the Rossby wave instability in thin disks using global 2D hydrodynamic simulations. The detailed linear theory of this nonaxisymmetric instability was developed earlier by Lovelace et al. and Li et al., who found that the instability can be excited when there is an extremum in the radial profile of an entropy-modified version of potential vorticity. The key questions we are addressing in this paper are: (1) What happens when the instability becomes nonlinear? Specifically, does it lead to vortex formation? (2) What is the detailed behavior of a vortex? (3) Can the instability sustain itself and can the vortex last a long time? Among various initial equilibria that we have examined, we generally find that there are three stages of the disk evolution: (1) The exponential growth of the initial small amplitude perturbations. This is in excellent agreement with the linear theory; (2) The production of large scale vortices and their interactions with the background flow, including shocks. Significant accretion is observed due to these vortices.(3) The coupling of Rossby waves/vortices with global spiral waves, which facilitates further accretion throughout the whole disk. Even after more than 20 revolutions at the radius of vortices, we find that the disk maintains a state that is populated with vortices, shocks, spiral waves/shocks, all of which transport angular momentum outwards. We elucidate the physics at each stage and show that there is an efficient outward angular momentum transport in stages (2) and (3) over most parts of the disk, with an equivalent Shakura-Sunyaev angular momentum transport parameter α in the range from 10 −4 to 10 −2 . By carefully analyzing the flow structure around a vortex, we show why such vortices prove to be almost ideal "units" in transporting angular momentum outwards, namely by positively correlating the radial and azimuthal velocity components. In converting the gravitational energy to the internal energy, we find some special cases in which entropy can remain the same while angular momentum is transported. This is different from the classical α disk model which results in the maximum dissipation (or entropy production). The dependence of the transport efficiency on various physical parameters are examined and effects of