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On the geometric origin of spurious waves in finite-volume discretizations of shallow water equations on triangular meshes. (English) Zbl 1453.65245
Summary: 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.
MSC:
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
65Z05 Applications to the sciences
Software:
MPAS-Ocean
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