A semi-Lagrangian Characteristic Mapping method for the solution of the tracer transport equations on the sphere is presented. The method solves for the solution operator of the equations by approximating the inverse of the diffeomorphism generated by a given velocity field. The evolution of any tracer and mass density can then be computed via pullback with this map. We present a spatial discretization of the manifold-valued map using a projection-based approach with spherical spline interpolation. The numerical scheme yields $C^1$ continuity for the map and global second-order accuracy for the solution of the tracer transport equations. Error estimates are provided and supported by convergence tests involving solid body rotation, moving vortices, deformational, and compressible flows. Additionally, we illustrate some features of computing the solution operator using a numerical mixing test and the transport of a fractal set in a complex flow environment.

翻译：提出了一种用于求解球面上示踪物输运方程的半拉格朗日特征映射方法。该方法通过逼近给定速度场所生成的微分同胚的逆算子来求解方程的解算子，从而可通过该映射的拉回计算任意示踪物及质量密度的演化。我们采用基于投影的球形样条插值方法，实现了流形值映射的空间离散化。该数值格式使映射具有$C^1$连续性，并保证示踪物输运方程解的二阶全局精度。通过涉及刚体旋转、移动涡旋、变形流及可压缩流的收敛性测试，验证了误差估计的可靠性。此外，我们还通过数值混合测试及复杂流环境中分形集的输运演示了解算子计算的部分特性。