Analysis of operator splitting for advection–diffusion–reaction problems from air pollution modelling

Operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations. It is, for example, standard practice in computational air pollution modelling where we encounter systems of three-dimensional, time-dependent partial differential equations of the advection–diffusion–reaction type. For such systems little attention has been devoted to the analysis of splitting and to the question why splitting can work so well. From the theoretical point of view, the success of splitting is primarily determined by the splitting error. This paper presents an analysis of operator splitting aimed at providing insight into the splitting error. Using the Lie operator formalism, a general expression is derived for a three-term Strang splitting in the pure initial value case. For a class of advection–diffusion–reaction problems the splitting error is analyzed in greater detail. A special case is discussed in which the splitting error can be reduced. Also some attention is paid to the use of operator splitting in initial boundary value problems.