One way to determine the fluid velocity, or concentration, at a time and point in space is to trace a particle pathline backwards through space and time to the starting location at the previous solution time. The departure point is the beginning point for the pathline from the previous solution time. Because the spatial distribution, or field for the constituent which moves with the fluid, is known at the previous solution time, the value for the previous solution time can be determined using spatial interpolation from the known values at the departure point. A new way to determine departure point location, called the semi-analytical upwind path line tracing (SUT) method, is presented that uses a semi-analytical solution for particle tracking rather than a discrete numerical solution like the Euler and Runge-Kutta methods. The semi-analytical solution provides a way to move entirely across a cell in one calculation while the discrete methods must divide the calculation into small pathline segments, or sub-calculations, for accuracy. Consequently, the SUT method has equivalent accuracy to discrete numerical solution approaches and can provide significantly improved computational efficiency for relatively long time step durations.

Abstract: A new method for departure point determination on Cartesian grids, the semi-analytical upwind path line tracing (SUT) method, is presented and compared to two typical departure point determination methods used in semi-Lagrangian advection schemes, the Euler method and the four-step Runge–Kutta method. Rigorous comparisons of the three methods were conducted for a severely curving hypothetical flow field and for advective transport in the rotation of a Gaussian concentration hill. The SUT method was shown to have equivalent accuracy to the Runge–Kutta method but with significantly improved computational efficiency. Depending on the case being simulated, the SUT method provides either far greater or equivalent computational efficiency and more certain accuracy than the Euler method.