In this subsection, we describe numerical formulations of the fourth-order MCV model for global transport computations on a Yin-Yang grid.
2.2.1 Governing equation
The two-dimensional spherical transport equation is written on the LAT-LON grid in conservative form as
,φ is the transported field,
is the Jacobian of the transformation, α is the radius of the Earth, λ and φ denote the longitude and latitude directions on a sphere, e and f are the flux components inλand φ directions,
is the flux vector, and
is the angular velocity on the LAT-LON grid.
2.2.2 Configuration of DOFs
A cubic spatial reconstruction over each control volume is required to construct a fourth-order MCV model. As a result, for single-cell reconstruction, 16 local DOFs should be defined within a 2D rectangular control volume. As shown in the computational space in Fig. 2, within control volume
, 16 local DOFs for model variable ψ(λ,φ,t) are defined at solution points Pi,j,m,n (denoted by solid circles) as
ψi,j,m,n (t)= ψ(λi,m,φj,n, t) (m=1…4, n=1…4), (3)
where i and j are indices of control volume, m and n are local indices of solution points Pi,j,m,n inλ andφ directions within each control volume,
are the locations of solution points.
The discretization formulations for these local DOFs, based on constraint conditions on multimoments, were developed in (Ii and Xiao, 2009). Hereafter we describe the implementation of a fourth-order MCV scheme for global transport computations on a Yin-Yang overset grid.
To construct a high-order model, the derivatives of the flux vectors [the second and third terms on the left-hand side of Eq, (2)] are first computed pointwise at the solution points, using the MCV scheme, to obtain semi-discrete equations. Then time integration is performed using a fourth-order Runge-Kutta method.
2.2.3 Spatial discretization of DOFs
On structured grids, a fully multidimensional MCV scheme can be constructed by efficiently implementing one-dimensional formulations in different directions. Without losing generality, we consider the computation of the derivative of flux component e with respect toλto discretize the governing equation in theλ-direction as
A similar procedure can be used along grid lines in the φ-direction by exchanging e with f and λ withφ.
For the fourth-order MCV model, equidistant local DOFs are defined within each control volume
. where 16 local DOFs are denoted by the solid circles. Within each control volume there are four rows (grid lines) in the φ-direction and four columns (grid lines) in the λ-direction.
Numerical formulations are described by considering the (nth=1…4) row of control volume Ci,j (Fig. 2). In the following discussion, we omit the subscripts j and n for sake of the conciseness. A cubic reconstruction polynomial can be built for any physical field using four local DOFs. For example, for model variableψ(λ,φj,n, t) we get
, where lm(λ) is a Lagrange basis function,
To build a MCV model, we first define the multimoment constraints, which are used to derive the spatial discretization equations for the DOFs. Three kinds of moments are adopted to construct a fourth-order scheme:Line-integrated averages (LIA) over the grid line are defined by
Point values (PVs) at the two endpoints of the grid line are defined by
First-order derivative values (DV) with respect toλat the center of the grid line are defined by
where subscript C denotes the center of the grid line.
The four DOFs and multimoment constraints are connected through the reconstruction polynomial Eq. (5)
Once we obtain formulations for updating the multimoment constraints in terms of the DOFs at the present time step, semidiscrete equations to predict the unknown DOFs can be derived by differentiating Eq. (9) with respect to time. We describe the discretization of the governing equations for the constraints as follows.
where ei1 and ei4 are pointwise values of flux component e at the two endpoints of the grid line.
Using the reconstruction profile Eq. (5), the transported field is continuous at cell interfaces. As a result, ei1 and ei4 can be directly computed by DOFs defined there.
The PV moment is updated by the differential-form Eq. (4) written at the two endpoints of the grid line. For example, at the left endpoint it is written as
where E is the spatial reconstruction for flux component e having the form in Eq. (5).
Finally, the time evolution of DOFs is computed by
To solve the problem with discontinuities or large gradients, a limiting projection using the TVB concept (Shu, 1987) is adopted in the MCV scheme to modify the DOFs to
where L(λ) is a linear reconstruction function written as
Here, b0 equals the value of LIA to assure conservation of the MCV model, and the slope b is determined using the Superbee limiter,
In the control volume where DOFs are modified by the above limiting projection, the continuity of transported field at cell interfaces cannot be preserved. Under this circumstance, we solve the Riemann problem to determine the pointwise value of flux component e. For example, ei1, mentioned above, is computed by
2.2.4 Time stepping
After accomplishing spatial discretization, the ordinary differential equation are obtained from
So far, we have developed a fourth-order transport model on a Yin-Yang grid using the MCV scheme, which is proposed under the Eulerian framework. In earlier work, we implemented numerical models on a Yin-Yang grid for a transport model (Li et al., 2006) and for a shallow-water model (Li et al., 2008) using semi-Lagrangian schemes. Although semi-Lagrangian schemes are widely used in GCMs, it is not easy to treat the integration of source terms along the trajectory with uniformly high-order accuracy. Therefore, for general systems of conservation laws, we prefer to adopt high-order schemes developed under the Eulerian framework.