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.

[1]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[2]  G. Burton.  Sobolev Spaces , 2013 .

[3]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[4]  P. Müller.  The Equations of Oceanic Motions , 2006 .

[5]  M. Webb,et al.  Quantification of modelling uncertainties in a large ensemble of climate change simulations , 2004, Nature.

[6]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[7]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[8]  Roger Temam,et al.  Models for the coupled atmosphere and ocean , 1993 .

[9]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[10]  Wolfgang Dahmen,et al.  Continuous refinement equations and subdivision , 1993, Adv. Comput. Math..

[11]  Y. Jaluria,et al.  An Introduction to Heat Transfer , 1950 .

[12]  M. Eldred,et al.  Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification , 2009 .

[13]  R. Temam,et al.  Models of the coupled atmosphere and ocean (CAO I). I , 1993 .

[14]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[15]  Roger Temam,et al.  Steady-state Navier–Stokes equations , 2001 .

[16]  Jonathan Rougier,et al.  Analyzing the Climate Sensitivity of the HadSM3 Climate Model Using Ensembles from Different but Related Experiments , 2009 .

[17]  Christophe Soligo,et al.  Origins and estimates of uncertainty in predictions of twenty-first century temperature rise , 2002 .

[18]  W. Washington,et al.  An Introduction to Three-Dimensional Climate Modeling , 1986 .

[19]  W. Collins,et al.  The Community Climate System Model Version 3 (CCSM3) , 2006 .

[20]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[21]  Mary F. Wheeler,et al.  Stochastic collocation and mixed finite elements for flow in porous media , 2008 .

[22]  Jonas Koko,et al.  Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids , 2006 .

[23]  G. J. Cooper,et al.  Additive Runge-Kutta methods for stiff ordinary differential equations , 1983 .

[24]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[25]  Roger Temam,et al.  Mathematical theory for the coupled atmosphere-ocean models (CAO III) , 1995 .

[26]  D. Rubenstein,et al.  Introduction to heat transfer , 2022 .

[27]  Matthew D. Collins,et al.  Towards quantifying uncertainty in transient climate change , 2006 .

[28]  M. Collins,et al.  The internal climate variability of HadCM3, a version of the Hadley Centre coupled model without flux adjustments , 2001 .

[29]  R. Temam,et al.  Numerical analysis of the coupled atmosphere-ocean models (CAO II). II , 1993 .

[30]  Leonard A. Smith,et al.  Uncertainty in predictions of the climate response to rising levels of greenhouse gases , 2005, Nature.

[31]  William J. Layton,et al.  Partitioned Time Stepping for a Parabolic Two Domain Problem , 2009, SIAM J. Numer. Anal..

[32]  S. Klein,et al.  GFDL's CM2 Global Coupled Climate Models. Part I: Formulation and Simulation Characteristics , 2006 .

[33]  Cosmin Safta,et al.  Uncertainty Quantification in the Presence of Limited Climate Model Data with Discontinuities , 2009, 2009 IEEE International Conference on Data Mining Workshops.

[34]  Gianluca Iaccarino,et al.  Large Eddy Simulations of Flow Around a Cylinder with Uncertain Wall Heating , 2009 .

[35]  William J. Layton,et al.  Decoupled Time Stepping Methods for Fluid-Fluid Interaction , 2012, SIAM J. Numer. Anal..

[36]  P. Jones,et al.  Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850 , 2006 .

[37]  Richard L. Smith,et al.  Quantifying Uncertainty in Projections of Regional Climate Change: A Bayesian Approach to the Analysis of Multimodel Ensembles , 2005 .

[38]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .