A standard test set for numerical approximations to the shallow water equations in spherical geometry

An unconditionally stable semi-Lagrangian method for the spherical atmospherical shallow water equations

“…In this section we show experiments performed with the new numerical scheme devised based on two test problems proposed by Williamson et al [9]. In the last few years, this set had been used for the validation of several shallow waters numerical models of the atmosphere (e.g.…”

“…If (0, 0 ) are NP co-ordinates in this system, the analytical solution to this test problem is given by We tested our model on a slightly rotated grid ( 0 = y/2− 0.05 rad) and on a full rotated grid ( 0 = 0.05 rad). This is Problem 2 described in Williamson et al [9].…”

“…6 in Williamson et al [9]). This is a frequently used meteorological test, but no analytical solution is known.…”

A radial basis Galerkin method for spherical surface Stokes equations

“…In our first example, u ∈ V ∩ H β (TS) is a smooth field with β > 2τ , and for the second nonsmooth case, 2 < β < 3 with β < τ . A standard approach to generate such benchmark vector fields in the atmospheric sciences, to demonstrate validity of numerical algorithms on the sphere, is through various stream functions [8]. For any scalar stream function u ∈ H β+1 (S) it is easy to verify that the vector field…”

“…Example 1 Motivated by the benchmark atmospheric low pressure flow field (popularly known as the Test Case 4 [5,8]) centred at x c = p(π/4, 0), for the first smooth test field, we choose u in (32) so that…”