Stability of algorithms for a two domain natural convection problem and observed model uncertainty

Two numerical algorithms are presented that couple a Boussinesq model of natural heat convection in two domains, motivated by the dynamic core of climate models. The first uses a monolithic coupling across the fluid–fluid interface. The second is a parallel implementation decoupled via a partitioned time stepping scheme with two-way communication. These new approaches are both proven to be unconditionally stable, a property not shared by traditional climate codes that use one-way coupling. Another critical property for a robust climate code is the reliability of the computation with respect to noise in unknown input parameters. This effect is measured by adding stochastic noise to two nonlinear coupling terms, involving momentum and heat transfer. Using an uncertainty quantification method known as stochastic collocation, we empirically measure the resulting variance in average surface temperature for each numerical model, including a third variant that employs one-way coupling for comparison. Throughout the duration of the simulation, we observe the smallest increase in variance using the monolithic algorithm and the largest increase using the one-way coupled algorithm. Partially supported by NSF grant DMS 0810385. J. M. Connors (B) · B. Ganis Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA e-mail: [email protected] B. Ganis e-mail: [email protected]