Application of the finite analytic numerical method to a flow-dependent variational data assimilation

An anisotropic diffusion filter can be used to model a flow-dependent background error covariance matrix, which can be achieved by solving the advection-diffusion equation. Because of the directionality of the advection term, the discrete method needs to be chosen very carefully. The finite analytic method is an alternative scheme to solve the advection-diffusion equation. As a combination of analytical and numerical methods, it not only has high calculation accuracy but also holds the characteristic of the auto upwind. To demonstrate its ability, the one-dimensional steady and unsteady advection-diffusion equation numerical examples are respectively solved by the finite analytic method. The more widely used upwind difference method is used as a control approach. The result indicates that the finite analytic method has higher accuracy than the upwind difference method. For the two-dimensional case, the finite analytic method still has a better performance. In the three-dimensional variational assimilation experiment, the finite analytic method can effectively improve analysis field accuracy, and its effect is significantly better than the upwind difference and the central difference method. Moreover, it is still a more effective solution method in the strong flow region where the advective-diffusion filter performs most prominently.