Numerical methods for multiphysics flow problems

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2016
Belenli Akbaş, Mine
In this dissertation, efficient and reliable numerical algorithms for approximating solutions of multiphysics flow problems are investigated by using numerical methods. The interaction of multiple physical processes makes the systems complex, and two fundamental difficulties arise when attempting to obtain numerical solutions of these problems: the need for algorithms that reduce the problems into smaller pieces in a stable and accurate way and for large (sometimes intractable) amount of computational resources to resolve all the physical scales. Although these two difficulties are often stated as separate issues, in practice they are quite related. The objective of this thesis is to advance state of the art in algorithms, and their better understanding through analysis, for two types of multiphysics problems: incompressible non-isothermal fluid flow, and magnetohydrodynamic flow. The first component of this thesis is to develop numerical algorithms that decouple the multiphysics systems of equations into smaller, easier to solve sub-problems. However, splitting up problems into components is well known to (sometimes dramatically) reduce accuracy and cause numerical instabilities. It will be rigorously proven that the decoupling algorithms proposed and studied herein are stable and accurate. Numerical tests are used to verify the stability and accuracy. The second component of the thesis is to construct the numerical scheme that use the Variational Multiscale (VMS) method to reduce the computational cost of these problems, by reducing the size of the smallest scale needing to be resolved. At the same time, the algorithm will decouple the VMS modeling/stabilization equations from the multiphysics system, and decouple the multiphysics system into its components. This thesis proposes such an efficient algorithm and rigorously proves it is stable and accurate, as well as giving guidance into picking the stabilization parameters. Numerical experiments verify the theoretical results, and reveal that the algorithm gives accurate solutions on coarse discretizations, i.e. with significantly less computational cost than the requirement to be resolved of the original (unstabilized) physical systems. Lastly, this thesis considers the notion of long-time stability for decoupling algorithms for multiphysics problems. It is quite common for stable numerical methods to be stable only over finite time intervals, and to produce numerical solutions that non-physically grow linearly or even exponentially with time, even when the true solution does not grow. Hence, it is desirable to use algorithms that are stable at all times, so that stability and accuracy can be preserved as long as possible in a numerical simulation. This thesis proves unconditional long time stability results for a particular class of linearized, second order methods for multiphysics problems and also for the usual incompressible Navier-Stokes equations.

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Citation Formats
M. Belenli Akbaş, “Numerical methods for multiphysics flow problems,” Ph.D. - Doctoral Program, Middle East Technical University, 2016.