Quantification of errors for operator-split advection–diffusion calculations

Multiphysics simulations frequently are composed from highly-optimized solvers for physical subprocesses through the use of temporal operator splitting. The subphysics are evolved sequentially, passing information between solver components as needed. It is often useful yet difficult to determine how the simulation error depends on the temporal splitting method, the time step size and the discretization parameters in the solver components. This paper proposes a framework to decompose the total error in a quantity of interest for an advection–diffusion simulation into two primary contributions: that due to the operator splitting, as implied by the structure of the code, and that due to the discretization errors from the component solvers. The method is applied to the advection–diffusion equation with boundaries. The advection and diffusion operators require separate boundary conditions upon splitting; the specification of these impacts the splitting error contribution. Computational examples demonstrate that the proposed method successfully identifies the splitting error contribution, including that induced by a suboptimal imposition of boundary conditions, and the decrease in the splitting error contribution in moving to a higher-order splitting method. The discretization error contribution is decomposed further into contributions attributed to the separate advection and diffusion solvers.