| Numerical simulation of different physical processes in different regions is one of the wide variety of real world applications. Many important applications such as coupled surface water groundwater flows require the accurate solution of multi-domain, multi-physics coupling of unobstructed flows with filtration or porous media flows. There are large advantages in efficiency, storage, accuracy and programmer effort in using partitioned methods build from components optimized for the individual sub-processes. On the other hand, the multidomain or multi-physical process describes different natures of the physical processes of each subproblem. They may require different meshes, time steps and methods. There is a natural desire to uncouple and solve such systems by solving each sub physics problem, to reduce the technical complexity and allow the use of optimized, legacy sub-problems' codes in fluid flow. Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear. Including a particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to many of the current models which use linear filters to select the eddies for damping.;The objective of the first part of this work is the development, analysis and validation of new partitioned algorithms for some coupled flow models, addressing some of the above problems. Particularly, this thesis studies: i) long time stability of four methods for splitting the evolutionary Stokes-Darcy problem into Stokes and Darcy sub problems, ii) analysis of a multi-rate splitting method for uncoupling evolutionary groundwater-surface water flows, and iii) a connection between filter stabilization and eddy viscosity models. For each problem, we give comprehensive analysis of stability and derive optimal error estimates of our proposed methods. Numerical experiments are performed to verify the theoretical results. |
