Reconstruction of vector fields for semi-Lagrangian advection on unstructured, staggered grids

Abstract

Applying the semi-Lagrangian method to discretize the advection of momentum eliminates the Courant number constraint associated with explicit Eulerian momentum advection in coastal ocean models. Key steps of the semi-Lagrangian method include calculating trajectories and interpolating the velocity vectors at the end of trajectories. In this work, we follow the linear and quadratic interpolation methods proposed by Walters et al. (2007) for field-scale simulations on unstructured, staggered grids and compare their performance using a backward-facing step test case and field-scale estuarine simulations. A series of methods to approximate the nodal and tangential velocities needed for the interpolation are evaluated and it is found that the methods based on the low-order Raviart–Thomas vector basis functions are more robust with respect to grid quality than the methods from Perot (2000) while overall they obtain similar accuracy. Over the range of different nodal and tangential velocities, the quadratic interpolation methods consistently exhibit higher accuracy than the linear interpolation methods. For the quadratic interpolation, the overall accuracy depends on the approximation of the tangential velocity. The backward-facing step test case indicates that the quadratic interpolation behaves like Eulerian central differencing or first-order upwinding, depending on the tangential approximations. The field-scale estuarine flow test case also shows general improvement for the velocity predictions and sharper gradients in the velocity field with the quadratic interpolation. The quadratic interpolations add less than 15% to the total computational time, and parallel implementation is relatively straightforward in complex geometries.