10. MLS-Based Selective Limiting for Shallow Waters Equations

Finite Volume Methods were successfully used in the last years to solve differential hyperbolic problems (Leveque, Finite-volume methods for hyperbolic problems. Cambridge University Press, Cambridge, 2005). Our research is focused here on the use of the Moving Least Squares (MLS) approximations for the development of a selective limiting technique to keep the accuracy of high-order methods in non-smooth flows. Following (Nogueira et al., Comput Methods Appl Mech Eng 199:2544–2558, 2010) we use the multiresolution properties of the MLS methodology and we define a shock-detection technique to act as a smoothness indicator. This sensor is used to detect shock waves present in the flow problem. The use of this technique combined with slope limiters improves the accuracy of the resulting TVD scheme. In this work we present the first results obtained with this technique applied to the resolution of the shallow waters equations with a high-order FV-MLS scheme (Cueto-Felgueroso et al., Comput Methods Appl Mech Eng 196:4712–4736, 2007). We present several 1D results and we compare them with those obtained with other high-order schemes.