Trajectory calculations for spherical geodesic grids in Cartesian space

This paper shows how to obtain accurate and efficient trajectory calculations for spherical geodesic grids in Cartesian space. Determination of the departure points is essential to characteristic-based methods that trace the value of a function to the foot of the characteristics and then either integrate or interpolate at this location. In this paper, the departure points are all computed in relation to the spherical geodesic grids that are composed of a disjoint set of unstructured equilateral triangles. Interpolating and noninterpolating trajectory calculation approaches are both illustrated and the accuracy of both methods are compared. The noninterpolating method of McGregor results in the most accurate trajectories. The challenge in using McGregor’s method on unstructured triangular grids lies in the computation of the derivatives required in the high-order terms of the Taylor series expansion. This paper extends McGregor’s method to unstructured triangular grids by describing an accurate and efficient method for constructing the derivatives in an element by element approach typical of finite element methods. An order of accuracy analysis reveals that these numerical derivatives are second-order accurate.