Finite-volume discretizations can be formulated on unstructured meshes composed of different polygons. A staggered cell-vertex finite-volume discretization of shallow water equations is analyzed on mixed meshes composed of triangles and quads. Although triangular meshes are most flexible geometrically, quads are more efficient numerically and do not support spurious inertial modes of triangular cell- vertex discretization. Mixed meshes composed of triangles and quads combine benefits of both. In particular, triangular transitional zones can be used to join quadrilateral meshes of differing resolution. Based on a set of examples involving shallow water equations, it is shown that mixed meshes offer a viable approach provided some background biharmonic viscosity (or the biharmonic filter) is added to stabilize the triangular part of the mesh.