The main objective of this thesis is to develop a numerical method for nonlinear partial differential equations (PDEs) which adapts the space-time computational mesh to the spacetime structure of the solution. Exploiting simultaneously the space and time localization of the solution, the method uses an optimal number of space-time degrees of freedom to provide an accurate and efficient solution to many problems in science and engineering.; In chapter 2, we review the adaptive wavelet collocation method (AWCM) for PDEs. In chapter 3, we develop a simultaneous space-time adaptive wavelet collocation method (STAWCM) that solves PDEs with local time steps, which controls the global space-time error. We develop a multi-level solver for nonlinear systems of equations in chapter 4 that combines the full approximation scheme (FAS) with wavelet compression. We verify the performance of the STAWCM on 1D+t-dimensional test problems in chapter 5, and on a 2D+t test problem in chapter 6. The STAWCM is used to study two-dimensional turbulence in chapter 7, where we investigate how the number of active space-time degrees of freedom N of a turbulent flow scales with Reynolds number Re (a non-intermittent estimate is N ∼ Re3/2). We simulate two-dimensional homogeneous isotropic turbulence for a range of Reynolds number 1 260 ≤ Re ≤ 40 400. Analysing this numerical data base, we estimate that N ∼ Re0.9 as Re → infinity.; The STAWCM has the following key properties: (i) control of the global space-time error by a specified tolerance, (ii) adaptive multi-level solver for a nonlinear system of algebraic equations, and (iii) optimal adaptive space-time mesh. We show that this technique is a promising candidate for accurate and efficient direct simulation of high Re turbulence. Finally in chapter 8, we discuss other applications of the STAWCM and future improvements to the method.
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