High-order conservative and oscillation-suppressing transport on irregular hexagonal grids

Abstract

A third-order numerical scheme was developed for 2D irregular hexagonal meshes for the advection problems in this study. The scheme is based on a multi-moment constrained finite-volume method (MCV) in Cartesian coordinates and entails the introduction of a general integration method over a hexagonal cell. Unlike in the conventional finite-volume method, various discrete moments, that is, point value and volume-integrated average, are adopted as computational constraints to achieve high-order computation. The high-order spatial reconstruction can therefore be built in a local space, which considerably reduces the stencil length. The numerical scheme is tested using various idealized experiments. Compared with the existing schemes, this scheme is demonstrated to be flexible for application in irregular hexagonal meshes without increasing cost or compromising on accuracy. The general integration formulation based on a third-order polynomial helps to expand the application to arbitrary hexagons that does not require the use of centroids as computational points or Voronoi tessellation. It is also convenient to define the orthogonal wind components in the Cartesian system to directly drive the atmospheric transport.

Appendix: Jacobian matrix J in triangle remapping

Fig. 10

General triangle remapped to an isosceles right triangle in a local coordinate space

Remapping of the general triangle to an isosceles right triangle is performed to facilitate the easy arrangement of the integral value of the prognostic variables over a triangle mesh. The key is determining the Jacobian matrix for the coordinate transformation. Figure 10 shows a schematic of the Jacobian operator that transforms an arbitrary triangle into an isosceles right triangle in the local coordinate space. Seven points (see Fig. 4) in the global coordinate system are remapped to $\phi _0(\frac{1}{3},\frac{1}{3})$, $\phi _1(0,1)$, $\phi _2(0,\frac{1}{2})$, and $\phi _3(0,0)$. $\phi _4(\frac{1}{2},0)$, $\phi _5(1,0)$, and $\phi _6(\frac{1}{2},\frac{1}{2})$ in the local coordinate system, respectively. The transformation of global coordinate to the local coordinate is then:

$$\begin{aligned}&x(\xi ,\eta )=\sum _{i=0}^{6} c_i(\xi ,\eta )x_i, \end{aligned}$$

(35)

$$\begin{aligned}&y(\xi ,\eta )=\sum _{i=0}^{6} c_i(\xi ,\eta )y_i, \end{aligned}$$

(36)

where $c_i(\xi ,\eta )$ is the same as that in (20), and denotes a set of basis functions of the Lagrange polynomial.

The Jacobian matrix (22) is expressed as:

$$\begin{aligned} J(\xi ,\eta )=\begin{vmatrix} \sum \limits _{i=0}^{6} \frac{\partial c_i(\xi ,\eta )}{\partial \xi }x_i~~\sum \limits _{i=0}^{6} \frac{\partial c_i(\xi ,\eta )}{\partial \eta }x_i\\ \sum \limits _{i=0}^{6} \frac{\partial c_i(\xi ,\eta )}{\partial \xi }y_i~~\sum \limits _{i=0}^{6} \frac{\partial c_i(\xi ,\eta )}{\partial \eta }y_i, \end{vmatrix}, \end{aligned}$$

(37)

where $\frac{\partial c_i(\xi ,\eta )}{\partial \xi }$, and $\frac{\partial c_i(\xi ,\eta )}{\partial \eta }$ are constants in the isosceles right triangle, as listed in Table 1. For example, J at $(\xi _6,\eta _6)$ is:

$$\begin{aligned}&J_6=J(\xi _6,\eta _6)=\sum \limits _{i=0}^{6} {\frac{\partial c_i(\xi ,\eta )}{\partial \xi }x_i}|_{\xi =\xi _6 ,\eta =\eta _6}\nonumber \\&\quad =-\frac{27}{4}x_0+\frac{1}{4}x_1+x_2+\frac{1}{4}x_3+x_4-\frac{3}{4}x_5+5x_6. \end{aligned}$$

(38)

Table 1 The elements of the Jacobian matrix

As shown in the left panel in Fig. 10, the locations of the middle points and centroid are then:

$$\begin{aligned} {\left\{ \begin{array}{ll} x_0=\frac{1}{3}(x_1+x_3+x_5), ~\\ x_2=\frac{1}{2}(x_1+x_3), ~\\ x_4=\frac{1}{2}(x_3+x_5), ~\\ x_6=\frac{1}{2}(x_1+x_5), \end{array}\right. } \end{aligned}$$

(39)

and

$$\begin{aligned} \begin{aligned}&\sum \limits _{i=0}^{6} \frac{\partial c_6(\xi ,\eta )}{\partial \xi }x_i=-\frac{27}{4}\frac{(x_1+x_3+x_5)}{3}+\frac{1}{4}x_1\\&\quad \quad +\frac{1}{2}(x_1+x_3)+\frac{1}{4}x_3+\frac{1}{2}(x_3+x_5)-\frac{3}{4}x_5+\frac{5}{2}(x_1+x_5)\\&\quad =x_1(-\frac{9}{4}+\frac{1}{4}+\frac{1}{2}+\frac{5}{2})\\&\quad +x_3(-\frac{9}{4}+\frac{1}{2}+\frac{1}{4}+\frac{1}{2})+x_5(-\frac{9}{4}+\frac{1}{2}-\frac{3}{4}+\frac{5}{2})\\&\quad =x_1-x_3 . \end{aligned} \end{aligned}$$

(40)

Similarly:

$$\begin{aligned} \begin{aligned} \sum \limits _{i=0}^{6} \frac{\partial c_6(\xi ,\eta )}{\partial \eta }x_i=x_5-x_3,\\ \sum \limits _{i=0}^{6} \frac{\partial c_6(\xi ,\eta )}{\partial \xi }y_i=y_1-y_3,\\ \sum \limits _{i=0}^{6} \frac{\partial c_6(\xi ,\eta )}{\partial \eta }y_i=y_5-y_3. \end{aligned} \end{aligned}$$

(41)

The Jacobian matrix (38) is then:

$$\begin{aligned} {\begin{aligned} J_6&=\begin{vmatrix} x_1-x_3~~x_5-x_3\\ \\ y_1-y_3~~y_5-y_3 \end{vmatrix}\\&=(x_1-x_3)(y_5-y_3)-(x_5-x_3)(y_1-y_3)\\&=2S, \end{aligned} } \end{aligned}$$

(42)

where $S =\frac{1}{2}[(x_1-x_3)(y_5-y_3)-(x_5-x_3)(y_1-y_3)]$ is the area of the triangle. $J_0, ..., J_5$ can be obtained in the same manner.