The error structure of the Douglas–Rachford splitting method for stiff linear problems

The Lie splitting algorithm is frequently used when splitting stiff ODEs or, more generally, dissipative evolution equations. It is unconditionally stable and is considered to be a robust choice of method in most settings. However, it possesses a rather unfavorable local error structure. This gives rise to severe order reductions if the evolution equation does not satisfy extra compatibility assumptions. To remedy the situation one can add correction-terms to the splitting scheme which, e.g., yields the first-order Douglas–Rachford (DR) scheme. In this paper we derive a rigorous error analysis in the setting of linear dissipative operators and inhomogeneous evolution equations. We also illustrate the order reduction of the Lie splitting, as well as the far superior performance of the DR splitting.