A nonlinear filtering algorithm for multi-dimensional finite element pollutant advection schemes

Abstract

A nonlinear filter is developed for use with finite element methods in solving the atmospheric diffusion equation. Usually, high-order accurate finite element methods lead to ripples near sharp gradients. This is an undesirable feature in air quality modeling. The filter eliminates these ripples by adding artificial diffusion along the direction of the streamlines. Since it is applied only in regions where the ripples are located, the accuracy of the solution is maintained. Here, the filter was used with the streamline upwind Petrov-Galerkin method. It displayed good performance characteristics in the standard rotating puff test, and in a new test where the angular velocity profile is parabolic. It did not cause excessive crosswind diffusion that was present in a previously developed two-dimensional filter, or Forester filters applied to one-dimensional, spatially split algorithms. The new test problem was designed specifically to show that the rotating puff test may not always be the appropriate test to evaluate the performance of transport schemes, especially those that split the horizontal transport into one-dimensional operators. As expected, a one-dimensional splitting scheme (Chapeau function) displayed worse performance than fully two-dimensional schemes under more severe conditions of the new test problem.