Date of Award

8-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Dr. Leo Rebholz, Committee Chair

Committee Member

Dr. Timo Heister, Co-chair

Committee Member

Dr. Hyesuk Lee

Committee Member

Dr. Qingshan Chen

Abstract

This dissertation studies efficient numerical methods for approximating solu-tions to viscous, incompressible, time-dependent magnetohydrodynamic (MHD) flows and computing MHD flows ensembles. Chapter 3 presents and analyzes a fully discrete, decoupled efficient algorithm for MHD flow that is based on the Els¨asser variable formulation, proves its uncondi-tional stability with respect to the timestep size, and proves its unconditional con-vergence. Numerical experiments are given which verify all predicted convergence rates of our analysis, show the results of the scheme on a set of channel flow problems match well the results found when the computation is done with MHD in primitive variables, and finally illustrate that the scheme performs well for channel flow over a step. In chapter 4, we propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep, yet maintains un-conditional stability with respect to timestep size. The scheme is optimally accu-rate in space, and behaves like second order in time in practice. The proposed method chooses θ ∈ [0, 1], dependent on the viscosity ν and magnetic diffusiv-ity νm, so that unconditionally stability is achieved, and gives temporal accuracy O(∆t2 + (1 − θ)|ν − νm|∆t). In practice, ν and νm are small, and so the method be-haves like second order. We show the θ-method provides excellent accuracy in cases

where usual BDF2 is unstable. Chapter 5 proposes an efficient algorithm and studies for computing flow en-sembles of incompressible MHD flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through an efficient algo-rithm that, at each time step, uses the same coefficient matrix for each of the J system solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Els¨asser variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments. This work concludes with chapter 6, which proposes, analyzes and tests high order algebraic splitting methods for MHD flows. The key idea is to applying Yosida-type algebraic splitting to the incremental part of the unknowns at each time step. This reduces the block Schur complement by decoupling it into two Navier-Stokes-type Schur complements, each of which is symmetric positive definite and the same at each time step. We prove the splitting is third order in ∆t, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, and as a finite element discretization of an approximation to the un-split discrete system. Numerical tests are given to illustrate the theory and show the effectiveness of the method. Finally, conclusions and future works are discussed in the final chapter.

Share

COinS