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Title:  Highorder multistep asynchronous splitting methods (MASM) for the numerical solution of multipletimescale ordinary differential equations 
Author(s):  Gudla, Pradeep K 
Director of Research:  West, Matthew 
Doctoral Committee Chair(s):  West, Matthew 
Doctoral Committee Member(s):  Salapaka, Srinivasa; Dullerud, Geir; Vakakis, Alexander; Kloeckner, Andreas 
Department / Program:  Mechanical Sci & Engineering 
Discipline:  Mechanical Engineering 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Stiff Ordinary Differential Equations 
Abstract:  Often one encounters dynamical systems containing a wide range of natural frequencies. These multipletimescale systems, which are modeled using stiff ordinary differential equations, are well known to present a significant challenge in obtaining a numerical solution efficiently. Multipletimescale systems can be broadly classified into two classes: (1) systems with wellseparate discrete time scales, such as molecular dynamic simulations and electrical networks, and (2) systems with a continuouslydistributed range of time scales, such as aerosol dynamics, multiscale structural dynamics and turbulent fluid flow. For the numerical simulation of systems with wellseparated discrete time scales one can frequently average over the fast time scales, either analytically or numerically. This results in effective models with only slower time scales and allows efficient numerical simulations with large timesteps. In cases where this is not possible, either due to system complexity or the fact that there is simply a wide range of timescales with no clear scale separation, such as the continuouslydistributed time scales systems, it has traditionally been necessary to simulate the entire system at the rate of the fastest timescale, which can be very expensive. To efficiently simulate multipletimescale systems, many researchers have developed multipletimestep numerical integration methods, where more than one timestep are used. These advance different components of the system forward in time at different rates, so that faster components can use small timesteps, while slower components can use large timesteps, resulting in lower computational cost. Most multipletimestep integrators only apply to systems with discrete time scales, where subcycling methods, mollified methods, and rRESPA are good examples. In addition, these methods which have several numerical timesteps require that timestep ratios be integer multiples of each other. In contrast, one family of multipletimestep methods does not attempt to enforce any such restrictions, namely asynchronous integrators. These methods have incommensurate timesteps, such that all system components never synchronize at common time instants. This feature allows some asynchronous methods to be efficiently applied to systems with continuouslydistributed time scales, where every time scale can have an appropriatelychosen numeral timestep. However, currently known asynchronous methods are at most secondorder accurate and are known to suffer from resonance instabilities, severely limiting their practical efficiency gain relative to their synchronous counterparts. In the present work, a new family of highorder Multistep Asynchronous Splitting Methods (MASM) is developed, based on a generalization of both classical linear multistep methods and the previouslyknown Asynchronous Splitting Methods (ASMs). These new methods compute highorder trajectory approximations using the history of system states and force vectors as for linear multistep methods, while at the same time allowing incommensurate timesteps to be used for different system components as in ASMs. This allows them to be both highorder and asynchronous, and means that they are applicable to systems with either discrete time scales or continuouslydistributed time scales. Consistency and convergence are established for these new highorder asynchronous methods both theoretically and via numerical simulations. For convergence, the only requirement is that the ratio of smallest to largest timestep remains bounded from above as the timesteps tend to zero. For a sufficiently regular ODE systems, an $m$step MASM can achieve m^th order accuracy, which is proven analytically and then validated using numerical experiments. Numerical simulations show that these methods can be substantially more efficient than their synchronous counterparts. Given that appropriate timesteps are chosen, the efficiency gain using MASM compared to synchronous multistep methods largely depends upon the force field splitting used. MASM is proven to be a stable method, provided it is convergent. The stability criterion also strongly depends upon the splitting of the force field chosen. In case of linear systems for which the Jacobian of the force vector is diagonalizable, the force vector splitting can be classified into asynchronous splitting, where each eigencomponent is lumped with one of the component force vector, and \emph{time scale splitting}, where an eigencomponent is split between two or more component force vectors. Any force vector splitting is in general a combination of asynchronous splitting and time scale splitting, where some eigencomponents are lumped with one of the component force vectors and other eigencomponents are split between different component force vectors. For synchronous splitting we prove a stability condition with a bound essentially the same as for the corresponding synchronous multistep method, while for time scale splitting we restrict the analysis to twocomponent systems and derive stability conditions for both conservative and nonconservative systems. Finally, we also present an efficient time step selection (TSS) strategy that can be employed while using MASM for numerically solving ODEs, yielding the TSSMASM method. This time step selection strategy is based on an optimal equidistribution principle, where component timesteps are chosen so that the contribution from each split force field towards the local discretization error is equal. The efficiency gain is system dependent and splitting dependent and is investigated numerically. For strongly coupled systems such as a multiscale spring mass damper, TSSMASM has approximately the same efficiency as synchronous multistep methods, while for weakly coupled systems such as aerosol condensation, TSSMASM is much more efficient. 
Issue Date:  20160418 
Type:  Text 
URI:  http://hdl.handle.net/2142/90556 
Rights Information:  Copyright 2016 Pradeep Kumar Gudla 
Date Available in IDEALS:  20160707 20181213 
Date Deposited:  201605 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois