Large time-stepping methods for higher order time-dependent evolutionequationsbyXiaoliang XieA dissertation submitted to the graduate facultyin partial fulfillment of the requirements for the degree ofDOCTOR OF PHILOSOPHYMajor: Applied mathematicsProgram of Study Committee:Hailiang Liu, Major ProfessorL. Steven HouPaul SacksJim W. EvansSunder SethuramanIowa State UniversityAmes, Iowa2008Copyrightc° Xiaoliang Xie, 2008. All rights reserved.iiDEDICATIONTo my dear Mother Ruilin Zhang and father XiXi XieiiiACKNOWLEDGEMENTSFirst and foremost, I would like to express my deepest gratitude to Professor Hailiang Liu,my academic advisor, for his guidance and support. Throughout my PhD period , he hasencouraged me many times to keep moving forward and to look for creative solutions, and healways provided valuable comments whenever needed. His keen scientific insights were indis-pensable for this dissertation and will always be a source of inspiration for my future career.Besides, I want to thank Dr.Liu for providing great help in the thesis revision, and helping mekeep faith in myself even at the most difficult moments. He could not even realize how much Ihave learned from him, besides math he always teaches me how to become a successful person.I feel really lucky to have had Dr.Hailiang Liu as my advisor.Also, I would like to thank my other committee members, Professors Paul Sacks, SunderSethuraman, L. Steven Hou and Jim Evans as a group.Besides, cooperation from Dr.Liu’s other students are also important to my work. I appre-ciate a lot all the helpful comments from Dr.Zhongming Wang and Dr.Haseena Ahmed.At last, I am profoundly grateful to all those involved in providing me tremendous supportthroughout my five years at Iowa State University.1CHAPTER 1. INTRODUCTIONRecently there has been significant research interest in the study of large time-steppingnumerical methods for different types of Partial Differential Equations arising in various ap-plications. Many of the underlying Partial Differential Equations are characterized by timedependence, high order derivatives and strong nonlinearity. Their solutions are expected topossess special properties such as pattern formation, which is not easily observed in a shortperio d of time. Therefore, in solving such PDEs, long time behavior of numerial solutions is acritical issue.We consider a class of PDEs of the form∂tu = L(u) + N(u), (1.1)where L = L(x, u, Du, D2u, ...) is a linear operator which contains high order spatial derivativesand N = N(x, u, Du, D2u, ...) is a nonlinear operator which may also contain high orderderivatives. The high order derivatives force small time step in explicit time-discretizationaccording to the CFL condition. Various large time-stepping numerical schemes have beendeveloped for these type of PDEs. The implicit-explicit scheme is to smartly combine theimplicit and explicit approximation for linear and nonlinear spatial derivative operators. Thestability property can b e found in [4]. It has been proposed for reaction-diffusion problems [30],Navier-Stokes equations [20], and the KdV equation [6, 22]. The integrating factor IF methodhas been developed by Trefethen [33], and Cox and Matthews [9]. The idea is to solve highorder linear operator exactly by making change of variables. An extension of implicit-explicitmethod which is called fast spectral algorithm has been developed by Fornberg and Driscoll[12] for purely dispersive equations and has been generalized by Driscoll [11] by using Runge-Kutta time-stepping. The main idea is to use different numerical scheme for the low, medium2and high wavenumbers separately. Another is the exponential time differencing (ETD) schemewhich has been developed by Cox and Matthews for stiff systems [9]. The ETD scheme applythe same integrating factor as in the IF approach, the difference in the integration is over asingle time step of length h. Later ETD schemes have been improved to fix the numericalinaccuracy due to cancellation errors, see [21].1.1 Goals and Main ResultsOur goal is to design a class of large time-stepping stable methods for different types ofhigher order PDEs. These will be developed gradually as the complexity of equations increases.Part I: Linear equations(N=0 in (1.1))Our key idea for design of large time-stepping numerical schemes is based on a precondi-tioning approximation∂tu²= φ, £φ = P (∂x)u, (1.2)where the preconditioning operator £(², D) is selected to make L(u) φ shares equal regularityof u. In order for the approximate system to be faithful we require:(i)Consistency lim²→0£(², D) = I;(ii)£(², D) is a linear operator;(iii)² > 0 is a parameter proportional to the time step to be determined.We summarize the choice of £ for three different types of linear operator L in Theorem(2.1.1).1) £(², ∂x) = I − ²P (∂x),² =∆t2for Re(P (iξ)) = 0, for all ξ ∈ R,2) £(², ∂x) = I − ²P (∂x) with∆t2≤ ² ≤ ∆t for Re(P (iξ) < 0, for all ξ ∈ R,3) £(², ∂x) = I − ²(P (∂x) −c) with∆t2≤ ² ≤ ∆t for Re(P (iξ)) ≤ c(c > 0), for all ξ ∈ R.Where P (iξ) is the Fourier transform of P (∂x). By choosing £ in this way, a semi-discretescheme (2.4) is stable in the sense of||un+1||2≤ ec∆t||un||2. (1.3)3We also discussed the effect of spatial discretization on the stability of the fully discretescheme, we define A to be the matrix derived from the discretization of P (∂x) to have thefollowing results:1) When P (iξ) ≤ 0 for ξ ∈ R and∆t2≤ ² ≤ ∆t, a symmetric negative definite matrix A willmake fully discretized scheme unconditionally stable in the sense of ||un+1|| ≤ ||un||.2) When P (iξ) is pure imaginary and ² =∆t2, a skew-Hermitian matrix A will make fullydiscretized scheme unconditionally stable in the sense of ||un+1|| = ||un||.Where A is the spatial discretization matrix corresponding to linear operator P (∂x).Our study is mainly focused on stability issues, and we know for linear equations, as longas we have a stable scheme it is also a convergent scheme due to the Lax Equivalence Theorem.By the study of the linear equations we build a solid foundation for nonlinear equations, wealways apply the result from this part for the nonlinear equations.Part II: Semi-linear equations. When both N and L exist and the high order deriva-tives appear on the linear operator and the nonlinear operator is of low order, we regroup theoperators and then split the linear and nonlinear