A Block-Jacobi Time-Spectral Method For Incompressible Flow. Alton James Luder III Ph.D. Thesis, University of Michigan, Ann Arbor, MI, March, 2013.
A Block-Jacobi Time-Spectral Method For Incompressible Flow [link]Paper  abstract   bibtex   
The Time-Spectral Method is a method for discretization of the time-derivative of periodic functions. This method is applied to Partial Differential Equations (PDEs), such as those encountered in Computational Fluid Dynamics (CFD). The Time- Spectral Method utilizes the assumption of periodicity to couple all the discrete time- levels in a way that dramatically improves accuracy, and therefore reduces the number of discrete time-levels dramatically. By using a block-Jacobi solver, this method can be implemented in a standard CFD library that can perform linear system solves on one time level at a time. This solver is under-relaxed to maintain diagonal dominance instead of using dual-time stepping as in previous implementations. The Time-Spectral Method is tested on the Linear Advection-Diffusion Equation as well as Burgers’ Equation. Both equations show that the Time-Spectral Method can be over an order of magnitude faster than second order Backward Difference Formula (BDF) time-marching. The Time-Spectral Method is extended for use with the incompressible Navier-Stokes Equations via a pressure projection method. This implementation of the Time-Spectral Method is tested on a backward facing step, a pitching foil, and a plunging foil. Both the backward facing step and the pitching foil can be five times faster than BDF. The plunging foil simulation only manages to be two and half times faster than BDF. The Time-Spectral Method is used for turbulent incompressible flow. Turbulence modeling is implemented with the Unsteady Reynolds Averaged Navier-Stokes (URANS) technique and the Spalart-Allmaras turbulence closure model. This is first tested on a pitching foil which showed very good agreement with a simulation using a highly resolved BDF time-marching scheme. Next this turbulent implementation is tested with a plunging foil. The results showed that, for highly resolved flows, the Time-Spectral Method has less problems due to pressure checker boarding instability than BDF. Both of these cases show nonlinear oscillations for Time-Spectral discretizations that are more finely discretized. The Time-Spectral Method is also compared to BDF for a propeller in shear flow. The Time-Spectral Method shows a four-fold computational cost reduction over the BDF.
@phdthesis{alton_james_luder_iii_block-jacobi_2013,
	address = {Ann Arbor, MI},
	title = {A {Block}-{Jacobi} {Time}-{Spectral} {Method} {For} {Incompressible} {Flow}},
	url = {http://hdl.handle.net/2027.42/99935},
	abstract = {The Time-Spectral Method is a method for discretization of the time-derivative of periodic functions. This method is applied to Partial Differential Equations (PDEs), such as those encountered in Computational Fluid Dynamics (CFD). The Time- Spectral Method utilizes the assumption of periodicity to couple all the discrete time- levels in a way that dramatically improves accuracy, and therefore reduces the number of discrete time-levels dramatically. By using a block-Jacobi solver, this method can be implemented in a standard CFD library that can perform linear system solves on one time level at a time. This solver is under-relaxed to maintain diagonal dominance instead of using dual-time stepping as in previous implementations. The Time-Spectral Method is tested on the Linear Advection-Diffusion Equation as well as Burgers’ Equation. Both equations show that the Time-Spectral Method can be over an order of magnitude faster than second order Backward Difference Formula (BDF) time-marching. The Time-Spectral Method is extended for use with the incompressible Navier-Stokes Equations via a pressure projection method. This implementation of the Time-Spectral Method is tested on a backward facing step, a pitching foil, and a plunging foil. Both the backward facing step and the pitching foil can be five times faster than BDF. The plunging foil simulation only manages to be two and half times faster than BDF. The Time-Spectral Method is used for turbulent incompressible flow. Turbulence modeling is implemented with the Unsteady Reynolds Averaged Navier-Stokes (URANS) technique and the Spalart-Allmaras turbulence closure model. This is first tested on a pitching foil which showed very good agreement with a simulation using a highly resolved BDF time-marching scheme. Next this turbulent implementation is tested with a plunging foil. The results showed that, for highly resolved flows, the Time-Spectral Method has less problems due to pressure checker boarding instability than BDF. Both of these cases show nonlinear oscillations for Time-Spectral discretizations that are more finely discretized. The Time-Spectral Method is also compared to BDF for a propeller in shear flow. The Time-Spectral Method shows a four-fold computational cost reduction over the BDF.},
	language = {en},
	school = {University of Michigan},
	author = {{Alton James Luder III}},
	month = mar,
	year = {2013},
	file = {aluder_1.pdf:/Users/jcoller/Zotero/storage/TTNNURMN/aluder_1.pdf:application/pdf}
}

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