Efficient Numerical Methods for Magnetohydrodynamic Flow

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

A Penalty-projection based Efficient and Accurate Stochastic Collocation Method for Magnetohydrodynamic Flows

We propose, analyze, and test a penalty projection-based efficient and accurate algorithm for the Uncertainty Quantification (UQ) of the time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated regimes. The algorithm uses the Els\"asser variables formulation and discrete Hodge decomposition to decouple the stochastic MHD system into four sub-problems (at each time-step for each realization) which are much easier to solve than solving the coupled saddle point problems. Each of the sub-problems is designed in a sophisticated way so that at each time-step the system matrix remains the same for all the realizations but with different right-hand-side vectors which allows saving a huge amount of computer memory and computational time. Moreover, the scheme is equipped with ensemble eddy-viscosity and grad-div stabilization terms. The stability of the algorithm is proven rigorously. We prove that the proposed scheme converges to an equivalent non-projection-based coupled MHD scheme for large grad-div stabilization parameter values. We examine how Stochastic Collocation Methods (SCMs) can be combined with the proposed penalty projection UQ algorithm. Finally, a series of numerical experiments are given which verify the predicted convergence rates, show the algorithm's performance on benchmark channel flow over a rectangular step, and a regularized lid-driven cavity problem with high random Reynolds number and magnetic Reynolds number.Comment: 28 pages, 13 figure

Commutation error in reduced order modeling of fluid flows

For reduced order models (ROMs) of fluid flows, we investigate theoretically and computationally whether differentiation and ROM spatial filtering commute, i.e., whether the commutation error (CE) is nonzero. We study the CE for the Laplacian and two ROM filters: the ROM projection and the ROM differential filter. Furthermore, when the CE is nonzero, we investigate whether it has any significant effect on ROMs that are constructed by using spatial filtering. As numerical tests, we use the Burgers equation with viscosities ν = 10− 1 and ν = 10− 3 and a 2D flow past a circular cylinder at Reynolds numbers Re = 100 and Re = 500. Our investigation (i) measures the size of the CE in these test problems and (ii) shows that the CE has a significant effect on ROM development for high viscosities, but not so much for low viscosities

High order algebraic splitting for magnetohydrodynamics simulation

WOS: 000400878000009This paper proposes, analyzes and tests high order algebraic splitting methods for magnetohydrodynamic (MHD) flows. The main idea is to apply, at each time step, Yosida-type algebraic splitting to a block saddle point problem that arises from a particular incremental formulation of MHD. By doing so, we dramatically reduce the complexity of the nonsymmetric block Schur complement by decoupling it into two Stokes-type Schur complements, each of which is symmetric positive definite and also is the same at each time step. We prove the splitting is 0(Delta t(3)) accurate, 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, but also in a finite element context that uses the natural spatial norms. Numerical tests are given to illustrate the theory and show the effectiveness of the method. (C) 2017 Elsevier B.V. All rights reserved.NSFNational Science Foundation (NSF) [DMS DMS1522191]Second, third and fourth authors were partially supported by NSF Grant DMS DMS1522191

How Are BMI, Nutrition, and Physical Exercise Related? An Application of Ordinal Logistic Regression

Background: This paper performs a detailed ordinal logistic regression study in an evaluation of a survey at a university in South Texas, USA. We show that, for categorical data in our case, ordinal logistic regression works well. Methods: The survey was designed according to the guidelines in diet and lifestyle from the American Heart Association and the United States Department of Agriculture and was sent out to all registered students at Texas A&M International University in Laredo, Texas. Data analysis included 601 students’ results from the survey. Data analysis was conducted in Rstudio. Results: The results showed that, compared with students who do not have enough whole grain food and exercise, those who have enough in both tend to have normal BMIs. As age increases, BMI tends to be out of the normal range. Conclusions: Because BMI in this research has three categories, applying an ordinal logistic regression model to describe the relationship between an ordered categorical response variable and more explanatory variables has several advantages compared with other models, such as the linear regression model

An Optimally Accurate Discrete Regularization for Second Order Timestepping Methods for Navier-Stokes Equations

We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier-Stokes equations (NSE). The method combines a linear treatment of the advection term, together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O(Δt2) accurate. The idea of \u27curvature stabilization\u27 is new to CFD and is intended as an improvement over the commonly used \u27speed stabilization\u27, which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows