Spectral-in-time methods for time-evolution partial differential equations

Abstract

In this thesis, we develop and analyze spectral-in-time methods to solve time-evolution partial differential equations (PDEs). To improve the computational complexity of solving the resulting linear system, we develop an algorithm using QR factorization in conjunction with the Woodbury matrix identity \cite{max1950inverting}. Taking advantage of the intrinsic structure of the spectral discretizations of PDE problems, this algorithm can reduce the computational complexity compared to conventional direct solvers. However, as the dimension of the problem increases, solving the problem by applying the spectral-in-time method globally in time has a long execution time and requires more storage availability. To deal with these challenges, we propose an approach that embeds the spectral-in-time method within a time-stepping framework, which is similar to the classical time-stepping methods. This approach preserves high precision while considerably improving the computational efficiency by decomposing the entire temporal domain into manageable subintervals. Moreover, we extend our method beyond linear PDEs by using Newton's method within the spectral-in-time framework for nonlinear PDEs. Instead of computing and storing the exact Jacobian matrices, we develop a final-time Jacobian matrix approach that generates much sparser Jacobian matrices. Through comprehensive numerical experiments, we show that this novel approach produces results indistinguishable in accuracy from the use of the exact Jacobian-based method with a slight reduction in the convergence of Newton's method.