Fourier Finite-Element Method with Linear Basis Functions on a Sphere: Application to Elliptic and Transport Equations

Abstract

Abstract The Fourier finite-element method (FFEM) on the sphere, which performs with an operation count of O(N2 log2N) for 2N × N grids in spherical coordinates, was developed using linear basis functions. Dependent field variables are expanded with the Fourier series in the longitude, and the Fourier coefficients are represented with a series of first-order finite elements. Different types of pole conditions were incorporated into the Fourier coefficients of the scalar and vector variables in order to avoid discontinuity at the poles. For the Laplacian operator, the linear element was defined as a function of the sine of latitude instead of the latitude. The FFEM was applied to the derivatives of the first- and second-order elliptic equations and the transport equations. The scale-selective high-order Laplacian-type filter was implemented as a hyperviscosity. For the first-order derivative the fourth-order convergence rate of the accuracy, as is expected from the theoretical analysis, was achieved. Elliptic equations were found to be solved accurately without pole discontinuity, and the convergence rate turned out to be second order. The cosine bell advection, time-differenced with a third-order Runge–Kutta method, showed that the squared-norm error convergence rate was slightly above second order. Both the Gaussian bell advection and the deformational flow produced the theoretical convergence rate of fourth order. The high-order filter was found to be effective in maintaining a quasi-uniform resolution over the sphere, and thus allowed a large time step size. Sensitivity experiments of cosine bell advection over the poles revealed that the CFL number, as defined with the maximum grid size on the global domain, can be taken to be as large as unity.