Recent developments in shock-capturing schemes

In this paper we review the development of the shock-capturing methodology, paying special attention to the increasing nonlinearity in its design and its relation to interpolation. It is well-known that high-order approximations to a discontinuous function generate spurious oscillations near the discontinuity (Gibbs phenomenon). Unlike standard finite-difference methods which use a fixed stencil, modern shock-capturing schemes use an adaptive stencil which is selected according to the local smoothness of the solution. Near discontinuities this technique automatically switches to one-sided approximations, thus avoiding the use of discontinuous data which brings about spurious oscillations. 'Research was supported by the National Aeronautics and Space Administration under NAS