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NIA Seminar by Hiroaki Nishikawa |
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Date: June 14, 2007
Time: 3:30pm
Location: NIA, Rm 137
Additional Information: Presentation (pdf)
Toward a Future Navier-Stokes Code: Exploiting Additional Degrees of Freedom
Hiroaki Nishikawa, University of Michigan
This talk will give an overview of some selected topics in Computational Fluid Dynamics (CFD), which are virtually unrelated but fall within the idea of exploiting additional degrees of freedom, and discuss a future that they imply. It begins with a residual-based adaptive grid method, in which the nodal coordinates of a computational grid is included as additional unkowns and they are adjusted to adapt themselves to the nodal solutions by minimizing the cell-residuals, creating a grid specially tailored to fit the physics of the governing equations (e.g., a characteristic grid for hyperbolic equations). Then, the talk goes on to describe the technique of local preconditioning, which adds to a steady equation an artificial time-derivative multiplied by a special matrix such that a steady state is reached as quickly as possible. In particular, a simplified theory for constructing such a matrix for general two-dimensional partial differential equations is described. Following brief presentions of some other topics ( decomposition for multigrid, adaptive quadrature, a rotated-hybrid Riemann solver), the talk finally introduces a new approach for diffusion problem. The new approach is based on an equivalent first-order hyperbolic system, with solution gradients included as additional unknowns. Because the system is hyperbolic, all techniques developed for hyperbolic systems, including those mentioned above, are now directly applicable to diffusion problems. There is no need any more to prepare another set of similar techniques for diffusion equations, and integrating a diffusion scheme with an advection scheme is now made natural and easy. The talk describes the basic idea of the new approach and its remarkable consequences that would bring a dramatic change in the way the diffusion equation is solved. Finally, the talk concludes with a remark on the form of a future Navier-Stokes code implied by the methods described.
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