The suite of Taylor–Galerkin class schemes for ice transport on sphere implemented by the INMOST package

Realizations of the numerical solution of the scalar transport equation on the sphere, written in divergent form, are presented. Various temporal discretizations are considered: the one-step Taylor–Galerkin method (TG2), the two-step Taylor–Galerkin method of the second (TTG2), third (TTG3), and fourth (TTG4) orders. The standard Finite-Element Galerkin method with linear basis functions on a triangle is applied as spatial discretization. The flux correction technique (FCT) is implemented. Test runs are carried out with different initial profiles: a function from C ∞ (Gaussian profile) and a discontinuous function (slotted cylinder). The profiles are advected by reversible, nondivergent velocity fields, therefore the initial distribution coincides with the final one. The case of a divergent velocity field is also considered to test the conservation and positivity properties of the schemes. It is demonstrated that TG2, TTG3, and TTG4 schemes with FCT applied give the best result for small Courant numbers, and TTG2, TTG4 are preferable in case of large Courant number. However, TTG2+FCT scheme has the worst stability. The use of FCT increases the integral errors, but ensures that the solution is positive with high accuracy. The implemented schemes are included in the dynamic core of a new sea ice model developed using the INMOST package. The acceleration of the parallel program and solution convergence with spatial resolution are demonstrated.

Verification of a 3rd Generation FEM Spectral Wave Model for Shallow and Deep Water Applications

This paper shows some results of the work currently carried out to improve the wave forecasting and hindcasting in oceanic and coastal regions. A new spectral wave model with a flexible numerical scheme using triangular elements to describe the model domain was developed by Hsu et al. (2002). This new spectral wave model called WWM (Wind Wave Model) is feasible for the spectral wave modeling of irregular coastlines and complicated bathymetries because of its numerical scheme. The Wave Action Equation is solved with the aid of the Fractional Step Method (Yanenko, 1971). The Integration in the spatial space is carried out with the Taylor-Galerkin Method and the terms describing depth and current induced refraction are integrated with the aid of Leonard’s (1979) TVD Ultimate Quickest scheme, which was already introduced in the WWIII (H. Tolman, 1991) for the same purpose. In three applications the wave model was verified against in-situ spectral measurements of directional and non-directional wave buoys. The results show that the new spectral wave model is capable of hindcasting the wave climate with a comparable accuracy like the SWAN model (Ris et al., 1998), though with a better efficiency since fewer nodes are necessary to resolve the model domain and the boundary conditions adequately.

A TIME ACCURATE PSEUDO-WAVELET SCHEME FOR TWO-DIMENSIONAL TURBULENCE

In this paper, we propose a wavelet-Taylor–Galerkin method for solving the two-dimensional Navier–Stokes equations. The discretization in time is performed before the spatial discretization by introducing second-order generalization of the standard time stepping schemes with the help of Taylor series expansion in time step. Wavelet-Taylor–Galerkin schemes taking advantage of the wavelet bases capabilities to compress both functions and operators are presented. Results for two-dimensional turbulence are shown.