Numerical analyses of operator-splitting algorithms for the two-dimensional advection-diffusion equation

Holly and Preissmann's (HP) scheme has been the basis of a large number of operator splitting algorithms for the solution of the advection-diffusion equation. However, these algorithms, including HP, are first-order accurate in time due to splitting errors. Error analyses of these algorithms, incorporating splitting error and errors resulting from numerical solutions of the split advection and diffusion equations, are lacking. In this paper, error analysis of a second-order accurate adaptation of the HP scheme (AHP) is presented for the two-dimensional advection-diffusion equation. A modified AHP scheme (MAHP) is suggested to remove the ad-hoc nature of boundary conditions during the diffusion step of computations in both HP and AHP. As boundary conditions specified for the advection-diffusion equation are not applicable to the split equations, second-order accurate boundary conditions for the split advection and diffusion equations are derived. An analysis of numerical dispersion and dissipation associated with the numerical procedure for the advection equation is presented. The analysis establishes a criterion so that computational errors are small in two-dimensional advection dominated transport problems. Several numerical examples are presented to verify the numerical analyses presented in the paper. In addition, a review of the current status of operator splitting algorithms for the advection-diffusion equation is presented. The objective of the review is to identify the issues that have not been addressed in the previous studies.