Differentiability of Slope Limiters on Unstructured Grids

Despite emerging alternatives such as essentially nonoscillatory schemes, the use of slope limiters re-mains a standard means of eliminating oscillations in the second-order accurate solution of hyperbolic equations. However, slope limiters for unstruc-tured grids are known to cause convergence prob-lems. We demonstrate that fully-implicit solutions us-ing Newton-Krylov methods are particularly affected. The root cause of this problem is shown to be the lack of differentiability of the limiting procedure. A mod-ication to the limiting procedure is introduced and shown to improve convergence.