Parallel Algorithms for Semi-Lagrangian Advection

Numerical time-step limitations associated with the explicit treatment of advection-dominated problems in computational uid dynamics are often relaxed by employing Eulerian-Lagrangian methods. These are also known as semi-Lagrangian methods in the atmospheric sciences. Such methods involve backward time integration of a characteristic equation to nd the departure point of a uid particle arriving at an Eulerian grid point. The value of an advected eld at the departure point is obtained by interpolation. Both the trajectory integration and repeated interpolation innuence accuracy. We compare the accuracy and performance of interpolation schemes based on piecewise cubic polynomials and cubic B-splines in the context of a distributed-memory, parallel computing environment. The computational cost and inter-processor communication requirements for both methods are reported. Spline interpolation has better conservation properties but requires the solution of a global linear system, initially appearing to hinder a distributed-memory implementation. The proposed parallel algorithm for multidimensional spline-interpolation has almost the same communication overhead as local piecewise polynomial interpolation. We also compare various techniques for tracking trajectories given diierent values for the Courant number. Large Courant numbers require a high-order ODE solver involving multiple interpolations of the velocity eld.