High order filtering methods for approximating hyperbolic systems of conservation laws

In the computation of discontinuous solutions of hyperbolic systems of conservation laws, the recently developed ENO (essentially non-oscillatory) schemes appear to be very useful. However, they are computationally costly compared to simple central difference methods. In this paper we develop a filtering method which uses simple central differencing of arbitrarily high order accuracy, except when a novel local test indicates the development of spurious oscillations. At these points, generally few in number, we use the full ENO apparatus, maintaining (in fact, improving) the high order of accuracy, but removing spurious oscillations. Numerical results indicate the success of the method. We obtain high order of accuracy in regions of smooth flow without spurious oscillations for a wide range of problems and a significant speed up of generally a factor of almost three over the full ENO method.