High Resolution Mars Surface

General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as smearing of the solution or spurious oscillations. Since publication of Godunov’s order barrier theorem, which proved that linear methods cannot provide non-oscillatory solutions higher than first order (Godunov 1954, Godunov 1959), these difficulties have attracted a lot of attention and a number of techniques have been developed that largely overcome these problems. To avoid spurious or non-physical oscillations where shocks are present, schemes that exhibit a Total Variation Diminishing (TVD) characteristic are especially attractive. Two techniques that are proving to be particularly effective are MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) a flux/slope limiter method (van Leer 1979, Hirsch 1990, Tannehill 1997, Laney 1998, Toro 1999) and the WENO (Weighted Essentially Non-Oscillatory) method (Shu 1998, Shu 2009). Both methods are usually referred to as high resolution schemes (see diagram).