# Large time-stepping methods for higher order time-dependent evolution equations

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Large time stepping methods for higher order time dependent evolution equations by Xiaoliang Xie A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major Applied mathematics Program of Study Committee Hailiang Liu Major Professor L Steven Hou Paul Sacks Jim W Evans Sunder Sethuraman Iowa State University Ames Iowa 2008 c Xiaoliang Xie 2008 All rights reserved Copyright ii DEDICATION To my dear Mother Ruilin Zhang and father XiXi Xie iii ACKNOWLEDGEMENTS First 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 has encouraged me many times to keep moving forward and to look for creative solutions and he always provided valuable comments whenever needed His keen scientific insights were indispensable 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 me keep faith in myself even at the most difficult moments He could not even realize how much I have 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 Sunder Sethuraman L Steven Hou and Jim Evans as a group Besides cooperation from Dr Liu s other students are also important to my work I appreciate 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 support throughout my five years at Iowa State University 1 CHAPTER 1 INTRODUCTION Recently there has been significant research interest in the study of large time stepping numerical methods for different types of Partial Differential Equations arising in various applications Many of the underlying Partial Differential Equations are characterized by time dependence high order derivatives and strong nonlinearity Their solutions are expected to possess special properties such as pattern formation which is not easily observed in a short period of time Therefore in solving such PDEs long time behavior of numerial solutions is a critical issue We consider a class of PDEs of the form t u L u N u 1 1 where L L x u Du D2 u is a linear operator which contains high order spatial derivatives and N N x u Du D2 u is a nonlinear operator which may also contain high order derivatives The high order derivatives force small time step in explicit time discretization according to the CFL condition Various large time stepping numerical schemes have been developed for these type of PDEs The implicit explicit scheme is to smartly combine the implicit and explicit approximation for linear and nonlinear spatial derivative operators The stability property can be 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 method has been developed by Trefethen 33 and Cox and Matthews 9 The idea is to solve high order linear operator exactly by making change of variables An extension of implicit explicit method 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 RungeKutta time stepping The main idea is to use different numerical scheme for the low medium 2 and high wavenumbers separately Another is the exponential time differencing ETD scheme which has been developed by Cox and Matthews for stiff systems 9 The ETD scheme apply the same integrating factor as in the IF approach the difference in the integration is over a single time step of length h Later ETD schemes have been improved to fix the numerical inaccuracy due to cancellation errors see 21 1 1 Goals and Main Results Our goal is to design a class of large time stepping stable methods for different types of higher 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 preconditioning approximation t u P x u 1 2 where the preconditioning operator D is selected to make L u shares equal regularity of u In order for the approximate system to be faithful we require i Consistency lim 0 D I ii D is a linear operator iii 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 x I P x t 2 2 x I P x with for Re P i 0 for all R t 2 t for Re P i 0 for all R 3 x I P x c with t 2 t for Re P i c c 0 for all R Where P i is the Fourier transform of P x By choosing in this way a semi discrete scheme 2 4 is stable in the sense of un 1 2 ec t un 2 1 3 3 We also discussed the effect of spatial discretization on the stability of the fully discrete scheme we define A to be the matrix derived from the discretization of P x to have the following results 1 When P i 0 for R and t 2 t a symmetric negative definite matrix A will make fully discretized scheme unconditionally stable in the sense of un 1 un 2 When P i is pure imaginary and t 2 a skew Hermitian matrix A will make fully discretized 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 long as 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 we always 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 derivatives appear on the linear operator and the nonlinear operator is of low order we regroup the operators and then split the linear and nonlinear part applying the result for linear operator L according to the discussion for linear equations and deal with the nonlinear term wisely to get large time stepping numerical schemes Different nonlinear operator N should be treated differently to get a stable numerical scheme We give three examples of nonlinear equations of this type the Korteweg de Vries the Swift Hohenberg equation and the