Robust explicit relaxation technique for solving the Green-Naghdi equations

Abstract

This paper revisits an original relaxation technique introduced in Favrie and Gavrilyuk [9] for solving the Green-Naghdi equations. We propose a version of that method and a space/time approximation thereof that is scale invariant. The approximation in time is explicit and the approximation in space uses a length scale for the relaxation that is proportional to the mesh size. The new method is compatible with dry states and is provably positivity preserving under the appropriate CFL condition. The method is numerically validated against manufactured solutions and is illustrated by comparison with experimental results.