Space-time Spectral Methods for Differential equations

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Mathematics, in particular, the area known as differential equations, is the language of science and engineering. The flow of blood in arteries and air over aircraft wings are explained by partial differential equations (PDEs). Unfortunately, exact solutions of these equations are rare. Scientists and engineers rely on numerical methods to solve these equations. Classical techniques can result in many millions of nonlinear equations to solve, taxing even the most powerful computers. Spectral methods solve time-independent PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is smooth. They require far fewer unknowns than other methods achieving comparable accuracy. My students and I have been studying a new class of spectral methods for time-dependent PDEs, called space-time spectral methods, that converge spectrally (i.e., exponentially) in both space and time for important linear PDEs. We have shown rigorously the spectral convergence, as well as condition number estimates of these methods. The latter quantifies the degree of difficulty of solving the problems, and are the most important objects of study from the point of view of numerical analysis. We have also demonstrated numerically that space-time spectral methods are effective for many classes of nonlinear PDEs, including some of the most important PDEs in applications. Currently, the most serious drawback of space-time spectral method is that all unknowns over all times must be solved simultaneously. This presents a serious difficulty for 3D problems and/or nonlinear problems, and is the main issue that will be addressed in this proposal. We propose new algorithms that can solve the PDEs on parallel computers to speed up the computations. A second objective of this proposal is a space-time ultra-spherical spectral method for linear PDEs. The ultra-spherical spectral method is a recent class of spectral methods that possesses lower memory requirement and is more stable than traditional spectral methods. It is one of the most promising methods for numerical PDEs. Another objective of this research is a space-time spectral method for delay differential equations which are used extensively to model population dynamics. The final objective is further progress on a 25-year old open problem on a property about the spectrum of a space-time spectral fourth derivative operator. Space-time spectral methods are robust and universal solvers for linear and nonlinear PDEs. They have been shown to be effective for many types of PDEs possessing different mathematical properties. We believe that any progress in accelerating the solution process of space-time spectral methods can have a significant impact on scientific computing, and will provide tools to assist scientists and engineers in their discovery of new phenomena and innovations. Finally this research will provide valuable theoretical and computational training for HQPs.