UC Irvine

This dissertation proposes numerical methods for the Euler and Navier-Stokes equations with spectral discretization in time and a fast space-time coupled LU-SGS (ST-LU-SGS) method for solving the resultant implicit equations. Firstly, the Fourier time spectral method is studied for periodic problems with test cases. The problem of non-symmetric solutions for symmetric periodic flow problems, caused by odd numbers of intervals in a period, is discovered and discussed in detail. The requirement of ensuring symmetric solution is proposed. In problems where frequency is not known a priori, a new frequency search approach based on Fourier analysis of the lift coefficient is proposed to work with the time spectral method. Computational results show that initial guesses of the frequency far away from the exact value can be used if the new approach is applied before employing a gradient based method. A new Chebyshev time spectral method is proposed to solve non-periodic unsteady problems and is validated by test cases. Computational results show that this method is very efficient in simulating both periodic and non-periodic unsteady flows, especially the non-periodic problems.

The use of Fourier or Chebyshev spectral discretization in time results in implicit equations in time marching. Explicit Runge-Kutta methods have often been used to solve such implicit system of equations through the use of the dual-time stepping algorithm. Such methods are, however, slow despite the use of acceleration schemes such as implicit residual smoothing and multigrid. We propose a new space-time LU-SGS (ST-LU-SGS) implicit scheme for both the Fourier and Chebyshev time spectral methods. In this scheme, the time domain is regarded as one additional dimension in space. Computational experiments show that this new scheme is faster than the explicit Runge-Kutta solver. For Navier-Stokes flow test cases, computations using the ST-LU-SGS implicit scheme is over ten times faster than the explicit Runge-Kutta solver. The ST-LU-SGS implicit scheme also works very well with the proposed frequency search approach. The ST-LU-SGS scheme is as efficient as the Block-Jacobi implicit algorithm and is more robust than the Block-Jacobi implicit algorithm. The proposed ST-LU-SGS scheme works for problems with either low frequency or high frequency while the Block-Jacobi implicit algorithm fails for high frequency flow problems.