Semi-Lagrangian algorithm for two-dimensional advection-diffusion equation on curvilinear coordinate meshes

A semi-Lagrangian method for the solution of the unsteady advection-diffusion equation in complex geometries is presented. The domain is discretized into a set of control volumes defined by an orthogonal, boundary-fitted coordinate system. A splitting strategy is employed that decouples the advective and diffusive processes. The advective update is realized by calculating back trajectories along characteristic lines and then, at the feet of these trajectories, the value of the transported field is calculated by interpolation. The interpolated values are calculated using a tensor product cubic spline function. The method is simple to program and produces results of high accuracy. Numerical errors are considerably lower than for conventional Eulerian schemes. In one-dimensional and rectangular two-dimensional domains, the present method gives comparable results to the well-known semi-Lagrangian scheme. However, the new method allows for greater flexibility in handling complex domains.