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Numerical study of nonlinear evolution equations, using compact differencing

Li, Linzhong; (1997) Numerical study of nonlinear evolution equations, using compact differencing. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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This thesis consists of eight chapters and four appendices. Chapter one is an introduction, mainly concerning numerical schemes but partly also the present context in fluid dynamics. In chapter two, compact difference schemes (CDS) are introduced and reviewed, and then an extension for upwinding CDS is described. In addition, comparision between two different kinds of upwinded CDS is made through numerical experiments for Burgers' equation. Chapter three explores the application of CDS to the KdV equation, and the stability, conservative and phase properties of the proposed scheme are studied. In chapter four, fluid-dynamical theory is discussed regarding the collapse of an unsteady interacting boundary layer and the development of shortened length and time scales, in the near-wall dynamics of internal or external transitional-turbulent boundary layers or during dynamic stall. This theoretical study yields for certain internal flows an extended KdV equation and for external or quasi-external flows an extended Benjamin-Ono equation, governing the unknown surface pressure. The rest of the thesis is devoted to the numerical study of these two classes of evolution equations. First, in chapter five the treatments of boundary conditions and of Cauchy principal-value integrals are discussed, applying asymptotic expansions and Taylor expansions, respectively. In addition, a transformation and an algorithm are formulated in this chapter, for effective computation. Then chapters six and seven focus on the numerical computation of the extended KdV and Benjamin-Ono equations with CDS, respectively. The computation is performed in both the physical and the transformed planes and the effect of centred and noncentred schemes is carefully examined. This part of the work also includes the numerical tracking of nonlinear wave packets, numerical capture of finite-time break-ups and the calculation of the growth rate for a rapid secondary instability. Finally, concluding remarks are given in chapter eight.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Numerical study of nonlinear evolution equations, using compact differencing
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Thesis digitised by ProQuest.
Keywords: Applied sciences; Evolution equations
URI: https://discovery.ucl.ac.uk/id/eprint/10097684
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