A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations

Abstract

We introduce $\Psi\mathrm{ec}$, a local spectral exterior calculus for the two-sphere $S^2$. $\Psi\mathrm{ec}$ 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 $\dot{H}^{-r+1}( \Omega_{\nu}^{r} , S^2 )$ 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, $\Psi\mathrm{ec}$ is tailored towards structure preserving discretizations that can adapt to solutions with varying regularity. The construction of $\Psi\mathrm{ec}$ 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 $\Psi\mathrm{ec}$ for numerical computations using the rotating shallow water equations. Our numerical results demonstrate that a $\Psi\mathrm{ec}$-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.