Çoklu fizik problemleri için sayısal metotlar.

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.Ph.D. - Doctoral Progra

Numerical Studies on a Second Order Explicitly Decoupled Variational Multiscale Method

Projection based variational multiscale (VMS) methods are a very successful technique in the numerical simulation of high Reynolds number flow problems using coarse discretizations. However, their implementation into an existing (legacy) codes can be very challenging in practice. We propose a second order variant of projection-based VMS method for non-isothermal flow problems. The method adds stabilization as a decoupled post-processing step for both velocity and temperature, and thus can be efficiently and easily used with existing codes. In this work, we propose the algorithm and give numerical results for convergence rates tests and coarse mesh simulation of Marsigli flow

NUMERICAL ANALYSIS AND TESTING OF A FULLY DISCRETE, DECOUPLED PENALTY-PROJECTION ALGORITHM FOR MHD IN ELSASSER VARIABLE

We consider a fully discrete, efficient algorithm for magnetohydrodynamic (MHD) flow that is based on the Elsasser variable formulation and a timestepping scheme that decouples the MHD system but still provides unconditional stability with respect to the timestep. We prove stability and optimal convergence of the scheme, and also connect the scheme to one based on handling each decoupled system with a penalty-projection method. Numerical experiments are given which verify all predicted convergence rates of our analysis on some analytical test problems, 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 variable, and finally show the scheme performs well on a channel flow over a step

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

The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes

Schroeder, Philipp W./0000-0001-7644-4693WOS: 000442638700037grad-div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad-div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad-div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix-Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings. (C) 2018 Elsevier B.V. All rights reserved.German Academic Exchange Service (DAAD)Deutscher Akademischer Austausch Dienst (DAAD); program "Research Grants for Doctoral Candidates and Young Academics and Scientists", 2017/18 [57299291]; National Science FoundationNational Science Foundation (NSF) [DMS1522191]; U.S. ArmyUnited States Department of Defense [65294-MA]The authors would especially like to thank Christoph Lehrenfeld for several related fruitful discussions on stabilization and hybridization and the invaluable help he provided in using the finite element library NGSolve in the context of this work. Mine Akbas acknowledges support from the German Academic Exchange Service (DAAD) with the program "Research Grants for Doctoral Candidates and Young Academics and Scientists", 2017/18 (57299291). The third author was supported by National Science Foundation grant DMS1522191 and U.S. Army grant 65294-MA

Re-examining the characteristics of pediatric multiple sclerosis in the era of antibody-associated demyelinating syndromes.

Background: The discovery of anti-myelin oligodendrocyte glycoprotein (MOG)-IgG and anti-aquaporin 4 (AQP4)-IgG and the observation on certain patients previously diagnosed with multiple sclerosis (MS) actually have an antibody-mediated disease mandated re-evaluation of pediatric MS series. Aim: To describe the characteristics of recent pediatric MS cases by age groups and compare with the cohort established before 2015. Method: Data of pediatric MS patients diagnosed between 2015 and 2021 were collected from 44 pediatric neurology centers across Turkiye. Clinical and paraclinical features were compared between patients with dis-ease onset before 12 years (earlier onset) and >= 12 years (later onset) as well as between our current (2015-2021) and previous (< 2015) cohorts. Results: A total of 634 children (456 girls) were enrolled, 89 (14%) were of earlier onset. The earlier-onset group had lower female/male ratio, more frequent initial diagnosis of acute disseminated encephalomyelitis (ADEM), more frequent brainstem symptoms, longer interval between the first two attacks, less frequent spinal cord involvement on magnetic resonance imaging (MRI), and lower prevalence of cerebrospinal fluid (CSF)-restricted oligoclonal bands (OCBs). The earlier-onset group was less likely to respond to initial disease-modifying treatments. Compared to our previous cohort, the current series had fewer patients with onset < 12 years, initial presentation with ADEM-like features, brainstem or cerebellar symptoms, seizures, and spinal lesions on MRI. The female/male ratio, the frequency of sensorial symptoms, and CSF-restricted OCBs were higher than reported in our previous cohort. Conclusion: Pediatric MS starting before 12 years was less common than reported previously, likely due to exclusion of patients with antibody-mediated diseases. The results underline the importance of antibody testing and indicate pediatric MS may be a more homogeneous disorder and more similar to adult-onset MS than previously thought