Accuracy of numerical methods for solving the advection–diffusion equation as applied to spore and insect dispersal

Abstract Three algorithms for solving a simplified 3-D advection–diffusion equation were compared as to their accuracy and speed in the context of insect and spore dispersal. The algorithms tested were the explicit central difference (ECD) method, the implicit Crank–Nicholson (ICN) method, and the implicit Chapeau function (ICF) method. The three algorithms were used only to simulate the diffusion process. A hold-release wind shifting method was developed to simulate the wind advection process, which shifts the concentration an integer number of grids and accumulates the remaining wind travel distance (which is less than the grid spacing) to the next time step. The test problem was the dispersal of a cloud of particles (originally in only one grid cell) in a 3-D space. The major criterion for testing the accuracy was R 2 , which represents the proportion of the total variation in particle distribution in all grid cells that is accounted for by the particle distribution through numerical solutions. Other criteria included total remaining mass, peak positive density, and largest negative density. High R 2 values were obtained for the ECD method with (Δ t K z )/(Δ z ) 2 ≤0.5 (Δ t =time step; K z =vertical eddy diffusion coefficient; Δ z =vertical grid spacing), and for the two implicit methods with Δ t K z /(Δ z ) 2 ≤5. The ICN method gave higher R 2 values than the ICF method when the concentration gradients were high, but its accuracy decreased more rapidly with the progress of time than the ICF method with a combination of a large grid spacing and a large time step. With very steep concentration gradients, the ICF method generated huge negative values, the ICN method generated negative values to a lesser extent, and the ECD method generated only small negative values. It was also found that good mass and/or peak preservation did not necessarily correspond to a higher R 2 value. Based on the R 2 value and the requirement for concentration positivity, for simulations with steep concentration gradients, the ECD method would be most appropriate, followed by the ICN method, and the ICF method would be least appropriate due to large negative values. For simulations with low concentration gradients, the ECD or ICF or ICN method could be used, but the ICN method would not be appropriate for use in a combination of a large time step and a large grid spacing. The results from this study could help selection and use of appropriate numerical methods in studying the spatial dynamics of spores and insects.