A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations
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We introduce Ψec, a local spectral exterior calculus for the two-sphere S^2. Ψec provides a discretization of Cartan's exterior calculus on S^2 formed by spherical differential r-form wavelets. These are well localized in space and frequency and provide (Stevenson) frames for the homogeneous Sobolev spaces Ḣ^-r+1( Ω_ν^r , S^2 ) of differential r-forms. At the same time, they satisfy important properties of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz decomposition. Through this, Ψec is tailored towards structure preserving discretizations that can adapt to solutions with varying regularity. The construction of Ψec is based on a novel spherical wavelet frame for L_2(S^2) that we obtain by introducing scalable reproducing kernel frames. These extend scalable frames to weighted sampling expansions and provide an alternative to quadrature rules for the discretization of needlet-like scale-discrete wavelets. We verify the practicality of Ψec for numerical computations using the rotating shallow water equations. Our numerical results demonstrate that a Ψec-based discretization of the equations attains accuracy comparable to those of spectral methods while using a representation that is well localized in space and frequency.
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