The inverse distance weighted interpolation applied to a particular form of the path tubes Method: Theory and computation for advection in incompressible flow

A simplified version of the inverse distance weighted interpolation formula was mathematically analyzed and employed in the context of a semi-Lagrangian scheme for the equation of advection. The scheme used is a particular formulation of Path Tubes method, a physically intuitive conservative method whose formulation is based on the theoretical foundations of the mechanics of continuous media, which uses the so-called Reynolds's transport theorem to establish judiciously its main property written on the basis of a conservative integral equation. The resulting algorithm is a five-point explicit semi-Lagrangian scheme for incompressible flow. From rigorous deductions based on the von Neumann stability analysis, it was proven that this explicit method is unconditionally stable. The present algorithm was evaluated using test problems that are prototypes of more sophisticated mathematical models commonly employed in the prediction of intensity variations of weather fronts. In addition, our methodology was compared against two other methods available in the literature. The proposed explicit semi-Lagrangian scheme was able to work with long time steps and proved to be accurate, non-oscillatory and non-diffusive.