A multi‐moment transport model on cubed‐sphere grid

A conservative, single‐cell‐based semi‐Lagrangian transport model is proposed in this paper. Using multi‐moment concept, an additional moment, i.e. volume‐integrated average (VIA), is treated as the model variable besides the point value (PV) updated in the traditional semi‐Lagrangian schemes. A quadratic interpolation function is constructed based on local degrees of freedom defined within each single cell. The PV moment is advanced by the semi‐Lagrangian formulation, whereas the VIA moment is updated by a finite volume formulation to rigorously ensure the numerical conservation. The numerical fluxes are computed from the PV moments defined along the boundary edges of the control volume. The scheme is extended to the spherical geometry through the application of the cubed‐sphere grid that eliminates the polar singularity in the conventional longitude/latitude coordinates by using the quasi‐uniform grid spacing covering the whole sphere. The single‐cell‐based scheme is well suited for the treatment of the connections between different patches. A simple quasi‐monotone limiter to the PV moment is applied to suppress non‐physical oscillations. The proposed scheme has been validated via representative benchmark tests and the performance is competitive to other existing transport schemes. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order element‐based Galerkin method for the barotropic vorticity equation

A high‐order element‐based Galerkin method is developed to solve the non‐divergent barotropic vorticity equation (BVE). The solution process involves solving a conservative transport equation for the vorticity fields and a Poisson equation for the stream function fields. The discontinuous Galerkin method is employed for solving the transport equation and a spectral element method (continuous Galerkin) is used for the Poisson equation. A third‐order strong stability preserving explicit Runge–Kutta scheme is used for time integration. A series of tests have been performed to validate the model, which include the evolution of an idealized tropical cyclone and interaction of dual vortices in close proximity. The numerical convergence study is performed by solving the BVE on the sphere where the analytic solution is known. The test results are consistent with physical observations, and the model exhibits exponential convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

A ‘well‐balanced’ finite volume scheme for blood flow simulation

We are interested in simulating blood flow in arteries with a one‐dimensional model. Thanks to recent developments in the analysis of hyperbolic system of conservation laws (in the Saint‐Venant shallow water equations context) we will perform a simple finite volume scheme. We focus on conservation properties of this scheme, which were not previously considered. To emphasize the necessity of this scheme, we present how a too simple numerical scheme may induce spurious flows when the basic static shape of the radius changes. On the contrary, the proposed scheme is ‘well‐balanced’: it preserves equilibria of Q = 0. Then examples of analytical or linearized solutions with and without viscous damping are presented to validate the calculations. The influence of abrupt change of basic radius is emphasized in the case of an aneurism. Copyright © 2012 John Wiley & Sons, Ltd.