Temporal discretisation errors in non-iterative split-operator approaches to solving chemical reaction groundwater transport models

Split-operator approaches are methods in widespread use for numerical solutions of reaction/transport problems. However, only a couple of analyses of truncation errors resulting from operator splitting have been published. A range of linear, single species models is analysed here, leading to generalisations of earlier findings, in addition to developing new results. Linear retardation, radioactive decay and interphase mass transfer are considered in detail. Operator-splitting errors for both the standard two-step and alternating two-step methods are presented, the latter approach being popular as a way to reduce errors due to operator splitting. An ambiguity in implementing two-step methods is revealed. The truncation error analysis also shows that the application of the usual alternating split-operator method to an equilibrium reaction (linear retardation) actually increases the (first order in time) truncation error relative to the standard method. However, for homogeneous boundary conditions, the same reaction can be solved to second-order accuracy if the model derived from taking the limit of a nonequilibrium, interphase mass transfer process is used. For the linear reactions considered here, the alternating two-step method is always second-order accurate in the temporal discretisation.