Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics

We present a non-diffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the Ultra-Bee limiter of [20], [24]. We prove for the Ultra-Bee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e. piecewise constant) initial data, and state a conjecture of non-diffusion at infinite time based on some local over-compressivity of the scheme for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.