The goal of this paper is to obtain a well-balanced, stable, fast, and robust HLLC-typeapproximate Riemann solver for a hyperbolic nonconservative PDE system arising in aturbidity current model. The main difficulties come from the nonconservative nature of thesystem. A general strategy to derive simple approximate Riemann solvers fornonconservative systems is introduced, which is applied to the turbidity current model toobtain two different HLLC solvers. Some results concerning the non-negativity preservingproperty of the corresponding numerical methods are presented. The numerical resultsprovided by the two HLLC solvers are compared between them and also with those obtainedwith a Roe-type method in a number of 1d and 2d test problems. This comparison shows that,while the quality of the numerical solutions is comparable, the computational cost of theHLLC solvers is lower, as only some partial information of the eigenstructure of thematrix system is needed.

An anisotropic solution adaptive method based on unstructured quadrilateral meshes for inviscid compressible flows is proposed. The data structure, the directional refinement and coarsening, including the method for initializing the refined new cells, for the anisotropic adaptive method are described. It provides efficient high resolution of flow features, which are aligned with the original quadrilateral mesh structures. Five different cases are provided to show that it could be used to resolve the anisotropic flow features and be applied to model the complex geometry as well as to keep a relative high order of accuracy on an efficient anisotropic mesh.