Error Estimation of Pseudospectral Method for Solving the Barotropic Vorticity Equation

The barotropic vorticity equation model is very important in the research of meteorological science and applied mathematics. Many scientists pay attention to the research of numerical methods of this equation. The early works were mainly concerned with finite difference methods [2, 4, 8, 9, 13]. The spectral method has a convergence rate of ”infinite” order, i.e., the error decays faster than algebraically when the solution is infinitely smooth. This method has become one of the most powerful tools for the numerical solution of nonlinear partial differential equations arising in the fluid dynamics [1, 5, 10]. Many authors provide various spectral schemes and analyze the errors. Usually, only nonlinear problems in Cartesian coordinates are considered. But in meteorological science and some other fields [7, 14], one also has to deal with the problem defined on the spherical surface. Since spectral methods usually entail too much computation work, sometimes it is even impossible to implement them strictly [3, 6]. The pseudospectral methods are preferred over spectral methods in practice, due to their numerical efficiency. However, there is a price to pay in the form of aliasing error which cause poor accuracy and instability in resolution demanding problems. Therefore, various techniques have already been developed for de-aliasing [11, 12]. In this paper, by taking the Barotropic Vorticity equation as an example, the pseudospectral method for solving partial differential equations on spherical surface is discussed. An interpolation procedure, which is different from that in the ordinary sense, is proposed. Based on such an interpolation, a pseudospectral scheme is constructed. Its generalized stability and convergence are analyzed rigorously.