On the geometric origin of spurious waves in finite-volume discretizations of shallow water equations on triangular meshes

Abstract

Computational wave branches are common to linearized shallow water equations discretized on triangular meshes. It is demonstrated that for standard finite-volume discretizations these branches can be traced back to the structure of the unit cell of triangular lattice, which includes two triangles with a common edge. Only subsets of similarly oriented triangles or edges possess the translational symmetry of unit cell. As a consequence, discrete degrees of freedom placed on triangles or edges are geometrically different, creating an internal structure inside unit cells. It implies a possibility of oscillations inside unit cells seen as computational branches in the framework of linearized shallow water equations, or as grid-scale noise generally.Adding dissipative operators based on smallest stencils to discretized equations is needed to control these oscillations in solutions. A review of several finite-volume discretization is presented with focus on computational branches and dissipative operators.