Journal of Computational Physics
Spectral transform solutions to the shallow water test set
Journal of Computational Physics
Journal of Computational Physics
The spectral element method for the shallow water equations on the sphere
Journal of Computational Physics
Wave propagation algorithms for multidimensional hyperbolic systems
Journal of Computational Physics
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Shallow water model on a modified icosahedral geodesic grid by using spring dynamics
Journal of Computational Physics
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations
Journal of Computational Physics
Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics
Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics
A fourth-order accurate local refinement method for Poisson's equation
Journal of Computational Physics
The NCAR Spectral Element Climate Dynamical Core: Semi-Implicit Eulerian Formulation
Journal of Scientific Computing
A sequel to AUSM, Part II: AUSM+-up for all speeds
Journal of Computational Physics
Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows
Journal of Computational Physics
A wave propagation method for hyperbolic systems on the sphere
Journal of Computational Physics
Finite-volume transport on various cubed-sphere grids
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Shallow water model on cubed-sphere by multi-moment finite volume method
Journal of Computational Physics
Journal of Computational Physics
An analysis of 1D finite-volume methods for geophysical problems on refined grids
Journal of Computational Physics
A discontinuous/continuous low order finite element shallow water model on the sphere
Journal of Computational Physics
Journal of Computational Physics
MCore: A non-hydrostatic atmospheric dynamical core utilizing high-order finite-volume methods
Journal of Computational Physics
Journal of Computational Physics
Hermitian Compact Interpolation on the Cubed-Sphere Grid
Journal of Scientific Computing
Hi-index | 31.47 |
This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM^+-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge-Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM^+-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.