전지구 순압모델의 이산화를 위한 후리에 유한 요소법의 연구

Abstract

Fourier-Finite Element Method(FFEM) with linear basis functions was developed for the partial differential equations describing one-layer atmospheric motions on the spherical surface. Dependent field variables are expanded with the Fourier series in the longitude, and the Fourier coefficients are represented with the series of linear finite elements. The pole conditions of the Fourier coefficients were applied to the scalar-, vector-, and the nonlinear flux variables, and also to their first-order derivatives. For the Laplacian operator, the linear element was defined as a function of the sine of latitude instead of the latitude, considering that the metric terms are expressed with the second-order polynomial. The scale-selective high-order Laplacian-type filter which consists of multiple Helmholtz equations was implemented as a hyper-viscosity. The FFEM was applied to the derivatives of the first- and second-order, the advection equation, deformational flow, and the shallow water equations. The numerical solutions of the differential equations even for the strong nonlinear flow passing over the poles were found stable and accurate by virtue of the conservation property of Galerkin procedure and the high-order filter. In terms of the derivatives the fourth-order convergence rate of the accuracy, as is expected from the theoretical analysis, was achieved for the linear elements used in the present study. The high-order filter, effectively dampening the unresolvable smaller-scales, was also shown to provide a quasi-uniform resolution, and thus allowing the large time-step size in spite of narrowing grid-size towards poles.